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8/19/2019 Limit and Its Rules calculus
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Limit and its Rules
Muhammad Nadeem
School of Electrical Engineering &Computer
Sciences
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8/19/2019 Limit and Its Rules calculus
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Concept of LimitLet ƒ( x ) be defined on an open
interval about x0 except possibly at x0 itself. If
ƒ ( x ) gets arbitrarily close to
k for all x sufficiently close
to
x0, we say that ƒ approaches the limit
k as x approaches x0.
k x f Lim x x
=→
)(0
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8/19/2019 Limit and Its Rules calculus
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y
The limit value does not depend on how the function is
defined:
1
1)(
2
−
−= x
x x f
2
y
=
≠−
−
=
1 1
1 1
1
)(
2
x
x x
x
xg
2
y
1)( += x xh
2
x11−
1
x11−
1
x11−
1
2)()()(111
===→→→
xh Lim xg Lim x f Lim x x x
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8/19/2019 Limit and Its Rules calculus
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y
A function may fail to have a Limit at a point in its domain
1
=
≠=
0 0
0 1
)(
x
x x xg
y y
≥
<=
1 1
1 0)(
x
x xU
>
≤
=0
1sin
0 0
)( x
x
x
xh
x0 0 0
x x
It jumps: The unitstep function U(x)has no limit asbecause its
values
jump at x=0.
It grows too large tohave a limit: g(x) has nolimit as x
approaches to 0because the values of g
grow arbitrarily large.
It oscillates too much to
have a limit: h(x) has nolimit as x approaches to 0because the
function’svalues oscillate between
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8/19/2019 Limit and Its Rules calculus
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Rules of Limit
( ) )()()()(
xg Lim x f Lim xg x f Limc xc xc x
→→→
±=±
)()()()(
xg Lim x f Lim xg x f Limc xc xc x
→→→
•=•
(1)
(2)
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)(
)(
)(
)(
xg Lim
x f Lim
xg
x f Lim
c x
c x
c x
→
→
→
=
)()(
x f Limk x f k Lim
c xc x →→ •=•
( ) nc x
n
c x x f Lim x f Lim
)()(
→→
=
(3)
(4)
(5)
0)( ≠cg
Rc f ∈)(
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8/19/2019 Limit and Its Rules calculus
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Limit of Polynomials
0
1
1..............)( a xa xa xP
n
n
n
n ++= −
−
Limit of a polynomial can be found by substitution:
Then
If
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0
1
1..............)()( acacacP xP Lim
n
n
n
nc x
++== −
−→
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8/19/2019 Limit and Its Rules calculus
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Example: Find the limit
234
2
−−−→ x
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8/19/2019 Limit and Its Rules calculus
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5)2()2(4)2(3)2(
)543(
234
234
2
−−=
+−−−−−+−=
+−−+−→
x x x x Lim x
17−=
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8/19/2019 Limit and Its Rules calculus
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2+ x
Example: Find the limit
6522 ++→ x xm
x
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8/19/2019 Limit and Its Rules calculus
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6)2(5)2(
22
)65(
)2(
65
2
2
2
2
2
22
++
+=
++
+
=++
+
→
→
→ x x Lim
x Lim
x x
x Lim
x
x
x
5
1
6104
4
=
++=
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8/19/2019 Limit and Its Rules calculus
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3− x
Example: Find the limit
3423 +−→ x xm x
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8/19/2019 Limit and Its Rules calculus
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)1)(3(
3
33
3
34
3
3
23
23
−−
−=
+−−
−=
+−
−
→
→→
x x
x Lim
x x x
x Lim
x x
x Lim
x
x x
2
1
)1(3
=
−=
→ x Lim x
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8/19/2019 Limit and Its Rules calculus
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x x −42
Example: Find the limit
xm
x −→ 24
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8/19/2019 Limit and Its Rules calculus
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2
)2)(2(
2
)4(
2
4
4
4
2
4
−
−+=
−
−=
−
−
→
→→
x
x x x Lim
x
x x Lim
x
x x Lim
x
x x
16
)22(4
)2(4
=
+=
+= → x x Lim x
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8/19/2019 Limit and Its Rules calculus
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1− x
Example: Find the limit
231 −+→ x x
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8/19/2019 Limit and Its Rules calculus
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)23)(1(
23
23
23
1
23
1
11
++−=
++
++∗
−+
−=
−+
−
→→
x x Lim
x
x
x
x Lim
x
x Lim
x x
4
)23(1
=
++=
−
→
x Lim
x
x
x
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8/19/2019 Limit and Its Rules calculus
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Sandwich Theorem
L xh Lim xg Limc xc x
==→→
)()(
Suppose that g( x) ≤ f ( x)
≤ h( x)for all x in some open intervalcontaining c,
except possibly at x = c itself. Suppose also that
Then
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xmc x
=→
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8/19/2019 Limit and Its Rules calculus
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Example: Suppose the following inequality holds for values ofx
close to 0;
2−
20
cos1
x
x Lim x
−
→
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22422
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8/19/2019 Limit and Its Rules calculus
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2
1
242
12
0=
−
→
x Lim x
2
1
2
1
0=
→ x
Lim
Since
2
1cos12
0
=−
→ x
x Lim x
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By sandwich theorem
