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8/12/2019 M53 Lec1.1 Limits-OneSided1
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Limit of a Function and
One-sided limits
Mathematics 53
Institute of Mathematics (UP Diliman)
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For today
1 Limit of a Function: An intuitive approach
2 Evaluating Limits
3 One-sided Limits
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For today
1 Limit of a Function: An intuitive approach
2 Evaluating Limits
3 One-sided Limits
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Introduction
Given a function f(x)anda
,
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Introduction
Given a function f(x)anda
,
what is the value of f atxneara,
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Introduction
Given a function f(x)anda
,
what is the value of f atxneara,
but not equal toa?
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Illustration 1
Consider f(x) =3x 1.
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 1
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
x f(x)
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
x f(x)
2
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
x f(x)
2 5
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
x f(x)
2 51.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
x f(x)
2 51.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
Based on the table, as x gets closer and closer to 1,
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Illustration 1
Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
x f(x)
2 51.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
Based on the table, as x gets closer and closer to 1, f(x)gets closer and closer
to2.
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Ill i 1
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Illustration 1
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
1 1 2 3
1
1
2
3
4
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Ill t ti 1
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Illustration 1
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
1 1 2 3
1
1
2
3
4
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Ill stration 1
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Illustration 1
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
2 5
1.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
2 5
1.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
2 5
1.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
2 5
1.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
2 5
1.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
2 5
1.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
1 1 2 3
1
1
2
3
4
Asxgets closer and closer to 1, f(x)gets closer and closer to 2.
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Illustration 2
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Illustration 2
Consider: g(x) = 3x2 4x+1
x 1
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Illustration 2
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Illustration 2
Consider: g(x) = 3x2 4x+1
x 1=
(3x 1)(x 1)x 1
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Illustration 2
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Illustration 2
Consider: g(x) = 3x2 4x+1
x 1=
(3x 1)(x 1)x 1
=3x 1, x = 1
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Illustration 2
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Consider: g(x) = 3x2 4x+1
x 1=
(3x 1)(x 1)x 1
=3x 1, x = 1
1 1 2 31
1
2
3
4
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Illustration 2
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Consider: g(x) = 3x2 4x+1
x 1=
(3x 1)(x 1)x 1
=3x 1, x = 1
1 1 2 31
1
2
3
4
Asxgets closer and closer to 1,
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Illustration 2
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Consider: g(x) = 3x2 4x+1
x 1=
(3x 1)(x 1)x 1
=3x 1, x = 1
1 1 2 31
1
2
3
4
Asxgets closer and closer to 1,
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Consider: g(x) = 3x2 4x+1
x 1=
(3x 1)(x 1)x 1
=3x 1, x = 1
1 1 2 31
1
2
3
4
Asxgets closer and closer to 1,
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Consider: g(x) = 3x2 4x+1
x 1=
(3x 1)(x 1)x 1
=3x 1, x = 1
1 1 2 31
1
2
3
4
Asxgets closer and closer to 1,
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Consider: g(x) = 3x2 4x+1
x 1=
(3x 1)(x 1)x 1
=3x 1, x = 1
1 1 2 31
1
2
3
4
Asxgets closer and closer to 1, g(x)gets closer and closer to 2.
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Illustration 3
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Consider: h(x) =
3x 1, x =10, x=1
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Illustration 3
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Consider: h(x) =
3x 1, x =10, x=1
1 1 2 31
1
2
3
4
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Illustration 3
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Consider: h(x) =
3x 1, x =10, x=1
1 1 2 31
1
2
3
4
Asxgets closer and closer to 1,
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Illustration 3
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Consider: h(x) =
3x 1, x =10, x=1
1 1 2 31
1
2
3
4
Asxgets closer and closer to 1,
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Illustration 3
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Consider: h(x) =
3x 1, x =10, x=1
1 1 2 31
1
2
3
4
Asxgets closer and closer to 1,
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Consider: h(x) =
3x 1, x =10, x=1
1 1 2 31
1
2
3
4
Asxgets closer and closer to 1,
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Consider: h(x) =
3x 1, x =10, x=1
1 1 2 31
1
2
3
4
Asxgets closer and closer to 1, h(x)gets closer and closer to 2.
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Limit
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Limit
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Intuitive Notion of a Limit
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Limit
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Intuitive Notion of a Limita ,L
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Limit
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Intuitive Notion of a Limita ,L f(x): function defined on some open interval containinga, except possibly at a
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Limit
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Intuitive Notion of a Limita ,L f(x): function defined on some open interval containinga, except possibly at a
The limit of f(x)asxapproachesa isL
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Limit
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Intuitive Notion of a Limita ,L f(x): function defined on some open interval containinga, except possibly at a
The limit of f(x)
asx
approachesa
isL
if the values of f(x)get closer and closer to Lasxassumes values getting closer
and closer toabut not reachinga.
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Limit
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Intuitive Notion of a Limita ,L f(x): function defined on some open interval containinga, except possibly at a
The limit of f(x)
asx
approachesa
isL
if the values of f(x)get closer and closer to Lasxassumes values getting closer
and closer toabut not reachinga.
Notation:limxa f(x) = L
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Examples
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f(x) = 3x 1
1 1 2 3
1
1
2
3
4
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Examples
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f(x) = 3x 1
1 1 2 3
1
1
2
3
4 lim
x1(3x 1)
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Examples
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f(x) = 3x 1
1 1 2 3
1
1
2
3
4 lim
x1(3x 1) =2
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Examples
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f(x) = 3x 1
1 1 2 3
1
1
2
3
4 lim
x1(3x 1) =2
Note: In this case, limx1
f(x)
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Examples
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f(x) = 3x 1
1 1 2 3
1
1
2
3
4 lim
x1(3x 1) =2
Note: In this case, limx1
f(x) = f(1).
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Examples
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g(x) = 3
x
2
4
x+1
x 1
1 1 2 31
1
2
3
4
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Examples
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g(x) = 3
x
2
4
x+1
x 1
1 1 2 31
1
2
3
4 limx1
3x2 4x+1x 1
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Examples
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g(x) = 3x2
4x+1
x 1
1 1 2 31
1
2
3
4 limx1
3x2 4x+1x 1 =2
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Examples
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g(x) = 3x2
4x+1
x 1
1 1 2 31
1
2
3
4 limx1
3x2 4x+1x 1 =2
Note: Thoughg(1)is undefined,limx1
g(x)exists.
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Examples
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h(x) =
3x
1, x
=1
0, x=1
1 1 2 31
1
2
3
4
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Examples
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h(x) =
3x
1, x
=1
0, x=1
1 1 2 31
1
2
3
4
limx1
h(x)
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Examples
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h(x) =
3x
1, x
=1
0, x=1
1 1 2 31
1
2
3
4
limx1
h(x) =2
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Examples
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h(x) =
3x
1, x
=1
0, x=1
1 1 2 31
1
2
3
4
limx1
h(x) =2
Note:h(1) = limx1 h(x).
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Some Remarks
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Remark
In finding limxa f(x):
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Some Remarks
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Remark
In finding limxa f(x):
We only need to consider values of x very close toabut not exactly at a.
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Some Remarks
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Remark
In finding limxa f(x):
We only need to consider values of x very close toabut not exactly at a.
Thus, limxa f(x)is NOT NECESSARILYthe same as f(a).
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Some Remarks
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Remark
In finding limxa f(x):
We only need to consider values of x very close toabut not exactly at a.
Thus, limxa f(x)is NOT NECESSARILYthe same as f(a).
We letxapproachafrom BOTH SIDES ofa.
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Some Remarks
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If f(x)
does not approach any
particular real number asx
approachesa, then we say
limxa f(x)does not exist (dne).
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Some Remarks
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If f(x)
does not approach any
particular real number asx
approachesa, then we say
limxa f(x)does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 39
Some Remarks
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If f(x)does not approach any
particular real number asx
approachesa, then we say
limxa f(x)does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
3 2 1 1 2 3
1
2
3
0
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 39
Some Remarks
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If f(x)does not approach any
particular real number asx
approachesa, then we say
limxa f(x)does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
3 2 1 1 2 3
1
2
3
0
limx0
H(x) = 0?
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 39
Some Remarks
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If f(x)does not approach any
particular real number asx
approachesa, then we say
limxa f(x)does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
3 2 1 1 2 3
1
2
3
0
limx0
H(x) = 0? No.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 39
Some Remarks
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If f(x)does not approach any
particular real number asx
approachesa, then we say
limxa f(x)does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
3 2 1 1 2 3
1
2
3
0
limx0
H(x) = 0? No.
limx0
H(x) = 1?
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 39
Some Remarks
8/12/2019 M53 Lec1.1 Limits-OneSided1
80/289
If f(x)does not approach any
particular real number asx
approachesa, then we say
limxa f(x)does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
3 2 1 1 2 3
1
2
3
0
limx0
H(x) = 0? No.
limx0
H(x) = 1? No.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 39
Some Remarks
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If f(x)does not approach any
particular real number asx
approachesa, then we say
limxa f(x)does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
3 2 1 1 2 3
1
2
3
0
limx0
H(x) = 0? No.
limx0
H(x) = 1? No.
limx0
H(x) dne
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 39
For today
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1 Limit of a Function: An intuitive approach
2 Evaluating Limits
3 One-sided Limits
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 16 / 39
Limit Theorems
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Theorem
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 39
Limit Theorems
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Theorem
If limxa f(x)exists, then it is unique.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 39
Limit Theorems
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Theorem
If limxa f(x)exists, then it is unique.
Ifc , then limxa c
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 39
Limit Theorems
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Theorem
If limxa f(x)exists, then it is unique.
Ifc , then limxa c=c.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 39
Limit Theorems
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Theorem
If limxa f(x)exists, then it is unique.
Ifc , then limxa c=c.limxa x
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 39
Limit Theorems
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Theorem
If limxa f(x)exists, then it is unique.
Ifc , then limxa c=c.limxa x=a
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 39
Limit Theorems
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Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
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Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)]
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
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Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
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Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
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Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
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Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa
[f(x)g(x)]
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
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Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa
[f(x)g(x)] = limxa
f(x) limxa
g(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
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Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa
[f(x)g(x)] = limxa
f(x) limxa
g(x) =L1 L2
Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 39
Limit Theorems
Th
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Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa
[f(x)g(x)] = limxa
f(x) limxa
g(x) =L1 L2limxa[c f(x)] =
Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 39
Limit Theorems
Th
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Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa
[f(x)g(x)] = limxa
f(x) limxa
g(x) =L1 L2limxa[c f(x)] =climxa f(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 39
Limit Theorems
Th
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Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa
[f(x)g(x)] = limxa
f(x) limxa
g(x) =L1 L2limxa[c f(x)] =climxa f(x) =cL1
Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 39
Limit Theorems
Theorem
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Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) =L1 L2
limxa[c f(x)] =climxa f(x) =cL1
limxa
f(x)
g(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 39
Limit Theorems
Theorem
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Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) =L1 L2
limxa[c f(x)] =climxa f(x) =cL1
limxa
f(x)
g(x) =
limxa f(x)
limxag(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
Theorem
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Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) =L1 L2
limxa[c f(x)] =climxa f(x) =cL1
limxa
f(x)
g(x) =
limxa f(x)
limxag(x)
= L1L2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
Theorem
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Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) =L1 L2
limxa[c f(x)] =climxa f(x) =cL1
limxa
f(x)
g(x) =
limxa f(x)
limxag(x)
= L1L2
, providedL2=0
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
Theorem
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Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) =L1 L2
limxa[c f(x)] =climxa f(x) =cL1
limxa
f(x)
g(x) =
limxa f(x)
limxag(x)
= L1L2
, providedL2=0
limxa (f(x))
n
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
Theorem
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Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) =L1 L2
limxa[c f(x)] =climxa f(x) =cL1
limxa
f(x)
g(x) =
limxa f(x)
limxag(x)
= L1L2
, providedL2=0
limxa (f(x))
n =
limxa f(x)
n
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Limit Theorems
Theorem
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Theorem
Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) =L1 L2
limxa[c f(x)] =climxa f(x) =cL1
limxa
f(x)
g(x) =
limxa f(x)
limxag(x)
= L1L2
, providedL2=0
limxa (f(x))
n =
limxa f(x)
n = (L1)n
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 39
Evaluate: limx1
(2x2 +3x 4)
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107/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4)
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108/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
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Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 +
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110/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x
8/12/2019 M53 Lec1.1 Limits-OneSided1
111/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x
8/12/2019 M53 Lec1.1 Limits-OneSided1
112/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
8/12/2019 M53 Lec1.1 Limits-OneSided1
113/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
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=2
limx1 x2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
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=2
limx1 x2
+3
limx1 x
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
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=2
limx1 x2
+3
limx1 x limx1 4
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
2
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=2
limx1 x2
+3
limx1 x limx1 4
=2
limx1
x
2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
2
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=2
limx1 x2
+3
limx1 x limx1 4
=2
limx1
x
2+3
limx1
x
lim
x14
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
2
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=2
limx1 x2
+3
limx1 x limx1 4
=2
limx1
x
2+3
limx1
x
lim
x14
=2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
2
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=2
limx1 x
+3
limx1 x limx1 4
=2
limx1
x
2+3
limx1
x
lim
x14
=2(
1)2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
2
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=2
limx1 x
+3
limx1 x limx1 4
=2
limx1
x
2+3
limx1
x
lim
x14
=2(
1)2 +3
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
2
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=2
limx1 x
+3
limx1 x limx1 4
=2
limx1
x
2+3
limx1
x
lim
x14
=2(
1)2 +3(
1)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
l2
l l
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=2
limx1 x
+3
limx1 x limx1 4
=2
limx1
x
2+3
limx1
x
lim
x14
=2(
1)2 +3(
1)
4
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
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124/289
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
2
li2
3
li
li 4
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=2
limx1 x
+3
limx1 x limx1 4
=2
limx1
x
2+3
limx1
x
lim
x14
=2(
1)2 +3(
1)
4
= 5
In general:
Remark
If f is a polynomial function, then limxa f(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx1
(2x2 +3x 4)
limx1
(2x2 +3x 4) = limx1
2x2 + limx1
3x limx1
4
2
li2
3
li
li 4
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=2
limx1 x
+3
limx1 x limx1 4
=2
limx1
x
2+3
limx1
x
lim
x14
=2(
1)2 +3(
1)
4
= 5
In general:
Remark
If f is a polynomial function, then limxa f(x) = f(a).
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
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127/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
li 4x3 + 3x2 x + 1
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128/289
limx2
4x3 +3x2 x+1x2 +2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
li 4x3
+ 3x2
x + 1lim
x 2(4x3 +3x2 x+1)
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129/289
limx2
4x +3x x+1x2 +2
= x2 lim
x2(x2 +2)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
li 4x3
+ 3x2
x + 1lim
x2(4x3 +3x2 x+1)
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130/289
limx2
4x +3x x+1x2 +2
= x2 lim
x2(x2 +2)
= 4(8)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
lim 4x3
+ 3x2
x + 1lim
x2(4x3 +3x2 x+1)
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limx2
4x +3x x+1x2 +2
= x 2 lim
x2(x2 +2)
= 4(8) +3(4)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
lim 4x3
+ 3x2
x + 1 =lim
x2(4x3 +3x2
x+1)
8/12/2019 M53 Lec1.1 Limits-OneSided1
132/289
limx2
4x +3x x+1x2 +2
= x 2 lim
x2(x2 +2)
= 4(8) +3(4) (2) +1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
lim 4x3
+ 3x2
x + 1 =lim
x2(4x3 +3x2
x+1)
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limx2
4x +3x x+1x2 +2
= x 2lim
x2(x2 +2)
= 4(8) +3(4) (2) +1
4+2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
lim 4x3
+ 3x2
x + 1 =lim
x2(4x3 +3x2
x+1)
8/12/2019 M53 Lec1.1 Limits-OneSided1
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limx2
4x +3x x+1x2 +2
=lim
x2(x2 +2)
= 4(8) +3(4) (2) +1
4+2
= 176
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
lim 4x3
+ 3x2
x + 1 =lim
x2(4x3 +3x2
x+1)
8/12/2019 M53 Lec1.1 Limits-OneSided1
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limx2
4x +3x x+1x2 +2
=lim
x2(x2 +2)
= 4(8) +3(4) (2) +1
4+2
= 176
Remark
If fis a rational function and f(a)is defined,
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Evaluate: limx2
4x3 +3x2 x+1x2 +2
lim 4x3
+ 3x2
x + 12
=lim
x2(4x3 +3x2
x+1)
2
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limx2
4x +3x x+1x2 +2
lim
x2(x2 +2)
= 4(8) +3(4) (2) +1
4+2
= 176
Remark
If fis a rational function and f(a)is defined, then lim
xaf(x) = f(a).
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxan
f(x) = n
limxa f(x),
8/12/2019 M53 Lec1.1 Limits-OneSided1
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f ( )
f ( ),
provided limxa f(x) > 0whenn is even.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxan
f(x) = n
limxa f(x),
8/12/2019 M53 Lec1.1 Limits-OneSided1
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f ( )
f ( ),
provided limxa f(x) > 0whenn is even.
limx3
3x
1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) =
nlimxa f(x)
,
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f ( )
f ( )
provided limxa f(x) > 0whenn is even.
limx3
3x
1= limx3
(3x
1)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) =
nlimxa f(x)
,
8/12/2019 M53 Lec1.1 Limits-OneSided1
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f ( )
f ( )
provided limxa f(x) > 0whenn is even.
limx3
3x
1= limx3
(3x
1) =
8
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) =
nlimxa f(x)
,
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f ( )
f ( )
provided limxa f(x) > 0whenn is even.
limx3
3x
1= limx3
(3x
1) =
8= 2
2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) =
nlimxa f(x)
,
8/12/2019 M53 Lec1.1 Limits-OneSided1
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( )
( )
provided limxa f(x) > 0whenn is even.
limx3
3x
1= limx3
(3x
1) =
8= 2
2
limx1
3
x+4
x 2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) =
nlimxa f(x)
,
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provided lim
xa f(x) > 0whenn is even.
limx3
3x
1= lim
x3(3x
1) =
8= 2
2
limx1
3
x+4
x 2 = 3
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) =
nlimxa f(x)
,
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provided lim
xa f(x) > 0whenn is even.
limx3
3x
1= lim
x3(3x
1) =
8= 2
2
limx1
3
x+4
x 2 = 3
1+41 2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf
(x
) = nlimxa f
(x
),
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provided lim
xa f(x) > 0whenn is even.
limx3
3x
1= lim
x3(3x
1) =
8= 2
2
limx1
3
x+4
x 2 = 3
1+41 2 = 1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) = nlimxa
f(x),
8/12/2019 M53 Lec1.1 Limits-OneSided1
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provided lim
xa f(x) > 0whenn is even.
lim
x3
3x
1= lim
x3(3x
1) =
8= 2
2
limx1
3
x+4
x 2 = 3
1+41 2 = 1
limx7/2
4
3 2x
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) = nlimxa
f(x),
8/12/2019 M53 Lec1.1 Limits-OneSided1
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provided lim
xa f(x) > 0whenn is even.
lim
x3
3x
1= lim
x3(3x
1) =
8= 2
2
limx1
3
x+4
x 2 = 3
1+41 2 = 1
limx7/2
4
3 2x dne
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) = nlimxa
f(x),
8/12/2019 M53 Lec1.1 Limits-OneSided1
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provided lim
xa f(x) > 0whenn is even.
lim
x3
3x
1= lim
x3(3x
1) =
8= 2
2
limx1
3
x+4
x 2 = 3
1+41 2 = 1
limx7/2
4
3 2x dne
limx2
x2 4
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) = nlimxa
f(x),
8/12/2019 M53 Lec1.1 Limits-OneSided1
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provided lim
xa f(x) > 0whenn is even.
lim
x3
3x
1= lim
x3(3x
1) =
8= 2
2
limx1
3
x+4
x 2 = 3
1+41 2 = 1
limx7/2
4
3 2x dne
limx2
x2 4=??
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Theorem
Suppose limxa f(x)exists andn . Then,
limxa
nf(x) = nlimxa
f(x),
8/12/2019 M53 Lec1.1 Limits-OneSided1
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provided lim
xa f(x) > 0whenn is even.
lim
x3
3x
1= lim
x3(3x
1) =
8= 2
2
limx1
3
x+4
x 2 = 3
1+41 2 = 1
limx7/2
4
3 2x dne
limx2
x2 4=?? (for now)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 39
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
8/12/2019 M53 Lec1.1 Limits-OneSided1
151/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
=
limx32x2
5x+1
x3 x+4 3
8/12/2019 M53 Lec1.1 Limits-OneSided1
152/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
=
limx32x2
5x+1
x3 x+4 3
8/12/2019 M53 Lec1.1 Limits-OneSided1
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=
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
=
limx32x2
5x+1
x3 x+4 3
8/12/2019 M53 Lec1.1 Limits-OneSided1
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=
limx3
2x2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
=
limx32x2
5x+1
x3 x+4 3
8/12/2019 M53 Lec1.1 Limits-OneSided1
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=
limx3
2x2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
=
limx32x2
5x+1
x3 x+4 3
8/12/2019 M53 Lec1.1 Limits-OneSided1
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=
limx3
2x2
limx3
(5x+1)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
=
limx32x2
5x+1
x3 x+4 3
8/12/2019 M53 Lec1.1 Limits-OneSided1
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=
limx3
2x2
limx3
(5x+1)
limx
3(x3 x+4)
3
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
=
limx32x2
5x+1
x3 x+4 3
3
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=
limx3
2x2
limx3
(5x+1)
limx
3(x3 x+4)
3
=
18
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
=
limx32x2
5x+1
x3 x+4 3
3
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=
limx3
2x2
limx3
(5x+1)
limx
3(x3 x+4)
3
=
18 4
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
=
limx32x2
5x+1
x3 x+4 3
3
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=
limx3
2x2
limx3
(5x+1)
limx
3(x3 x+4)
3
=
18 4
28
3
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
Evaluate: limx3
2x2 5x+1
x3 x+4
3
limx32x2
5x+1
x3 x+4 3
=
limx32x2
5x+1
x3 x+4 3
3
8/12/2019 M53 Lec1.1 Limits-OneSided1
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=
limx3
2x2
limx3
(5x+1)
limx
3(x3 x+4)
3
=
18 4
28
3
= 1
8
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
8/12/2019 M53 Lec1.1 Limits-OneSided1
162/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
8/12/2019 M53 Lec1.1 Limits-OneSided1
163/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x+1
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x1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x+1
=0
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x1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x+1
=0and lim
x1(x 1)
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x1
x1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x+1
=0and lim
x1(x 1) = 0.
8/12/2019 M53 Lec1.1 Limits-OneSided1
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x1
x1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x+1
=0and lim
x1(x 1) = 0.
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x x
But whenx =1, 3x2 4x+1
x
1 =
(3x 1)(x 1)x
1 =3x 1.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x+1
=0and lim
x1(x 1) = 0.
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But whenx =1, 3x2 4x+1
x
1 =
(3x 1)(x 1)x
1 =3x 1.
Since we are just taking the limit asx 1,
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
8/12/2019 M53 Lec1.1 Limits-OneSided1
170/289
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x+1
=0and lim
x1(x 1) = 0.
8/12/2019 M53 Lec1.1 Limits-OneSided1
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But whenx =1, 3x2 4x+1
x
1 =
(3x 1)(x 1)x
1 =3x 1.
Since we are just taking the limit asx 1,
limx1
3x2 4x+1x 1 = limx1(3x 1)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Consider: g(x) = 3x2 4x+1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x+1
=0and lim
x1(x 1) = 0.
8/12/2019 M53 Lec1.1 Limits-OneSided1
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But whenx =1, 3x2 4x+1
x
1 =
(3x 1)(x 1)x
1 =3x 1.
Since we are just taking the limit asx 1,
limx1
3x2 4x+1x 1 = limx1(3x 1) =2.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 39
Definition
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 39
Definition
If limxa f(x) =0and limxag(x) = 0
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 39
Definition
If limxa f(x) =0and limxag(x) = 0then
lim
xa
f(x)
g(x)
is called anindeterminate form of type 0
0.
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yp0
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 39
Definition
If limxa f(x) =0and limxag(x) = 0then
lim
xa
f(x)
g(x)
is called anindeterminate form of type 0
0.
8/12/2019 M53 Lec1.1 Limits-OneSided1
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yp0
Remarks:
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 39
Definition
If limxa f(x) =0and limxag(x) = 0then
limx
a
f(x)
g(x)
is called anindeterminate form of type 0
0.
8/12/2019 M53 Lec1.1 Limits-OneSided1
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0
Remarks:
1 If f(a) = 0and g(a) =0, then f(a)g(a)
is undefined!
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 39
Definition
If limxa f(x) =0and limxag(x) = 0then
limx
a
f(x)
g(x)
is called anindeterminate form of type 0
0.
8/12/2019 M53 Lec1.1 Limits-OneSided1
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0
Remarks:
1 If f(a) = 0and g(a) =0, then f(a)g(a)
is undefined!
2 The limit above MAY or MAY NOT exist.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 39
Definition
If limxa f(x) =0and limxag(x) = 0then
limx
a
f(x)
g(x)
is called anindeterminate form of type 0
0.
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0
Remarks:
1 If f(a) = 0and g(a) =0, then f(a)g(a)
is undefined!
2 The limit above MAY or MAY NOT exist.
3 Some techniques used in evaluating such limits are:
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 39
Definition
If limxa f(x) =0and limxag(x) = 0then
limx
a
f(x)
g(x)
is called anindeterminate form of type 0
0.
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0
Remarks:
1 If f(a) = 0and g(a) =0, then f(a)g(a)
is undefined!
2 The limit above MAY or MAY NOT exist.
3 Some techniques used in evaluating such limits are:
FactoringRationalization
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 39
Examples
Evaluate: lim
x1
x2 +2x+1
x+1
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181/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 39
Examples
Evaluate: lim
x1
x2 +2x+1
x+1
0
0
8/12/2019 M53 Lec1.1 Limits-OneSided1
182/289
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 39
Examples
Evaluate: lim
x1
x2 +2x+1
x+1
0
0
limx 1
x2 +2x+1
x + 1
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183/289
x1 x+1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 39
Examples
Evaluate: lim
x1
x2 +2x+1
x+1
0
0lim
x 1x2 +2x+1
x + 1 = lim
x 1
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184/289
x1 x+1 x1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 39
8/12/2019 M53 Lec1.1 Limits-OneSided1
185/289
Examples
Evaluate: lim
x1
x2 +2x+1
x+1
0
0lim
x1x2 +2x+1
x + 1 = lim
x1(x+1)2
x + 1
8/12/2019 M53 Lec1.1 Limits-OneSided1
186/289
x 1 x+1 x 1 x+1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 39
Examples
Evaluate: lim
x1
x2 +2x+1
x+1
0
0lim
x1x2 +2x+1
x+1 = lim
x1(x+1)2
x+1
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x 1 x + 1 x 1 x + 1
= limx
1(x+1)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 39
Examples
Evaluate: limx
1
x2 +2x+1
x+1
0
0lim
x1x2 +2x+1
x+1 = lim
x1(x+1)2
x+1
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+ +
= limx
1(x+1)
= (1+1)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 39
Examples
Evaluate: limx
1
x2 +2x+1
x+1
0
0lim
x1x2 +2x+1
x+1 = lim
x1(x+1)2
x+1
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= limx
1(x+1)
= (1+1)
= 0
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 39
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Examples
Evaluate: limx
2
x3 +8
x2
4
0
0lim
x2x3 +8
x2 4
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 39
Examples
Evaluate: limx
2
x3 +8
x2
4
0
0lim
x2x3 +8
x2 4 = limx2
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 39
Examples
Evaluate: limx
2
x3 +8
x2
4
0
0lim
x2x3 +8
x2 4 = limx2(x+2)(x2 2x+4)
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 39
Examples
Evaluate: limx
2
x3 +8
x2
4
0
0lim
x2x3 +8
x2 4 = limx2(x+2)(x2 2x+4)
(x+2)(x 2)
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 39
Examples
Evaluate: limx
2
x3 +8
x2
4
0
0lim
x2x3 +8
x2 4 = limx2(x+2)(x2 2x+4)
(x+2)(x 2)2 2 4
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= limx
2
x2 2x+4x
2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 39
Examples
Evaluate: limx
2
x3 +8
x2
4
0
0lim
x2x3 +8
x2 4 = limx2(x+2)(x2 2x+4)
(x+2)(x 2)2 2 4
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= limx
2
x2 2x+4x
2
= 4+4+4
2 2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 39
Examples
Evaluate: limx
2
x3 +8
x2
4
0
0lim
x2x3 +8
x2 4 = limx2(x+2)(x2 2x+4)
(x+2)(x 2)2 2 + 4
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= limx
2
x2 2x+4x
2
= 4+4+4
2 2= 3
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 39
Examples
Evaluate: limx4
x2 162x
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
Examples
Evaluate: limx4
x2 162x
0
0
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2+
x
2+x
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2+
x
2+x
= limx4
(x2 16)(2+x)
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2+
x
2+x
= limx4
(x2 16)(2+x)4 x
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
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Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2+
x
2+x
= limx4
(x2 16)(2+x)4 x
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= limx4
(x 4)(x+4)(2+x)4 x
= limx4
[(x+4)(2+x)]
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2+
x
2+x
= limx4
(x2 16)(2+x)4 x
( )( )(
)
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= limx4
(x 4)(x+4)(2+x)4 x
= limx4
[(x+4)(2+x)]
= (8)(4)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2+
x
2+x
= limx4
(x2 16)(2+x)4 x
( )( )(
)
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= limx4
(x 4)(x+4)(2+x)4 x
= limx4
[(x+4)(2+x)]
= (8)(4)
= 32
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+2 3
x
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+2 3
x+4
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+2 3
x+43x2 +2 3
x+4
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+2 3
x+43x2 +2 3
x+4
= limx8
x 8
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
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Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+2 3
x+43x2 +2 3
x+4
= limx8
x 8(x 8)(x+1)( 3
x2 +2 3
x+4)
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+2 3
x+43x2 +2 3
x+4
= limx8
x 8(x 8)(x+1)( 3
x2 +2 3
x+4)
1
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= lim
x81
(x+1)( 3x2 +2 3x+4)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+2 3
x+43x2 +2 3
x+4
= limx8
x 8(x 8)(x+1)( 3
x2 +2 3
x+4)
1
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= lim
x81
(x+1)( 3x2 +2 3x+4)=
1
9(4+4+4)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+2 3
x+43x2 +2 3
x+4
= limx8
x 8(x 8)(x+1)( 3
x2 +2 3
x+4)
1
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= lim
x81
(x+1)( 3x2 +2 3x+4)=
1
9(4+4+4)
= 1
108
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 39
For today
1 Limit of a Function: An intuitive approach
2 Evaluating Limits
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3 One-sided Limits
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 29 / 39
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 39
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
Asx 1, the value of f(x)dependson whetherx < 1or x > 1.
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 39
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
Asx 1, the value of f(x)dependson whetherx < 1or x > 1. 4 3 2 1 1 2 3
1
2
3
4
0
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3
2
1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 39
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
Asx 1, the value of f(x)dependson whetherx < 1or x > 1. 4 3 2 1 1 2 3
1
1
2
3
4
0
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3
2
1
Asx approaches1 through values less than1,
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 39
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
Asx 1, the value of f(x)dependson whetherx < 1or x > 1. 4 3 2 1 1 2 3
1
1
2
3
4
0
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3
2
1
Asx approaches1 through values less than1, f(x)approaches 2.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 39
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
Asx 1, the value of f(x)dependson whetherx < 1or x > 1. 4 3 2 1 1 2 3
1
1
2
3
4
0
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3
2
1
Asx approaches1 through values less than1, f(x)approaches 2.Asx approaches1 through values greater than1,
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 39
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
Asx 1, the value of f(x)dependson whetherx < 1or x > 1. 4 3 2 1 1 2 3
1
1
2
3
4
0
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3
2
1
Asx approaches1 through values less than1, f(x)approaches 2.Asx approaches1 through values greater than1, f(x)approaches1.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 39
Illustration 5
Consider: g(x) =
x
2 1 1 2 3
1
2
0
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1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 31 / 39
Illustration 5
Consider: g(x) =
x
2 1 1 2 3
1
2
0
( )
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1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 31 / 39
Illustration 5
Consider: g(x) =
x
2 1 1 2 3
1
2
0
( )
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1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 31 / 39
Illustration 5
Consider: g(x) =
x
2 1 1 2 3
1
2
0
( )
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1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 31 / 39
Illustration 5
Consider: g(x) =
x
2 1 1 2 3
1
2
0
( )
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1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 31 / 39
Illustration 5
Consider: g(x) =
x
2 1 1 2 3
1
2
0
( )
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1
Since there is no open interval Icontaining0 such thatg(x)is defined onI,
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 31 / 39
Illustration 5
Consider: g(x) =
x
2 1 1 2 3
1
2
0
( )
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1
Since there is no open interval Icontaining0 such thatg(x)is defined onI, we
cannot letxapproach0 from both sides.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 31 / 39
Illustration 5
Consider: g(x) =
x
2 1 1 2 3
1
2
0
( )
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1
Since there is no open interval Icontaining0 such thatg(x)is defined onI, we
cannot letxapproach0 from both sides.
But we can say something about the values of g(x)as xgets closer and closer to0from the right of 0.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 31 / 39
One-sided Limits
Intuitive Definition
The
limit of f(x)asx approachesa from the left is L
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 32 / 39
One-sided Limits
Intuitive Definition
The
limit of f(x)asx approachesa from the left is L
if the values of f(x)get closer and closer to Las the values of xget closer and
closer toa, but are less thana.
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 32 / 39
One-sided Limits
Intuitive Definition
The
limit of f(x)asx approachesa from the left is L
if the values of f(x)get closer and closer to Las the values of xget closer and
closer toa, but are less thana.
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Notation:
limxa
f(x) = L
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 32 / 39
One-sided Limits
Intuitive Definition
The
limit of f(x)asxapproachesafrom the right is L
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 33 / 39
One-sided Limits
Intuitive Definition
The
limit of f(x)asxapproachesafrom the right is L
if t