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7/29/2019 Continuity stage (2)
1/28
Facebook page : https://www.facebook.com/MrSherifYehiaAlMaraghy
Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Continuitystage (2)-33-
x 1lim f ( x )
x 2 x 2
x 2
At x 2 :
2 1 1 x 1 1f ( 2 ) and lim f ( x ) lim
2 1 3 x 1 3
As lim f ( x ) f ( 2 ) ,Thus f is Continuous at x 2
At x -1 :
2f ( 1 ) , So , The function f ( x ) is undefined at x -1 .
0
Thus f ( x ) is discountinuos at x - 1
2x : x 2let f(x) Then Discuss the Continuity at x 2
4 : x 2
x 2 x 2
x 2
At x 2 : f ( 2 ) 2 2 0 and lim f ( x ) lim x 2 2 2 0
So , As lim f ( x ) f ( 2 ) ,Thus f is Continuous at x 2
f ( 2 ) is not defined f ( x ) is not Continous at x 2
2nd
stage : Continuity of a function at a point
The function Fis said to be continuous at a pointx = a, if three conditions are satisfied :
(1) F(a) is well defined (2)x alim f(x) exists
(3)
x alim f(x) f(a)
Note that :
If any one of these three conditions is not satisfied , then the function is said to be
discontinuous at x a .
----------------------------------------------------------------------------------------- --------------Simple Example
Discuss the continuity of the function : f ( x ) x 3 at x 1
Answerf(1) = 1+ 3 = 4 And
x 1 x 1lim f ( x ) lim x 3 1 3 4
So , f(1) = Then f(x) is Continous at x = 1 .
----------------------------------------------------------------------------------------- --------------Example (1)
x 1Discuss the continuity of the function : f ( x ) at x -1 and x 2
x 1
Answer
----------------------------------------------------------------------------------------- --------------Example (2)
Answer
----------------------------------------------------------------------------------------- --------------Example (3)
Discuss the continuity of the function : f ( x ) x 2 at x 2
Answer
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Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Continuitystage (2)-3-
x 3x
f ( x ) f ( 3 ) x 1 f ( 3 ) 2x 2
x 3
x 3
x 3
x 3
f ( 3 ) 3 1 4
for lim f(x):
left limit : f( 3 ) lim 2x 2 2( 3 ) 2 4
Right limit : f( 3 ) lim x 1 3 1 4
f( 3 ) f( 3 ) f( 3 ) f ( 3 ) lim f(x) 4
f ( x ) is Continous at x 3
4
x 1x
f ( x ) f (1 ) x 3 f (1 ) x 1 2
Example (4)
x 1 : x 3let f(x) Then Discuss the Continuity at x 3
2x 2 : x 3
Answer
----------------------------------------------------------------------------------------- --------------Example (5)
Discuss the continuity of the function f ( x ) x 1 2 at x 1. Answer
x 1 : x 1 x 1 2 : x 1x 1 f ( x )
-( x 1) : x 1 - x 1 2 : x 1
x 1 : x 1f ( x )
- x 3 : x 1
x 1
x 1
x 1
x 1
f (1) 1 1 2
for lim f(x):
left limit : f( 1 ) lim x 3 1 3 2
Right limit : f( 1 ) lim x 1 1 1 2
f( 1 ) f( 1 ) f( 1) f (1) lim f(x) 2
f ( x ) is Continous at x 1
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Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Continuitystage (2)-3-
x2
x
f ( x ) f Sinx Cos x2
x2
x2
x 2
x2
x2
for lim f(x):
left limit : f( ) lim Sin x Cos x 12
Right limit : f( ) lim 2 Cos2x 1
2
f( ) f( ) lim f(x) 12 2
f ( ) lim f(x) f ( x ) is continous at x2 2
1 f 2 Cos2x2
Example (6)
sin x cos x : x2
If f(x) Then Discuss the Continuity at x22 cos 2x : x
2
Answer
f ( ) Sin Cos 12 2 2
----------------------------------------------------------------------------------------- --------------
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Calculus 3rdsecondary Continuitystage (2)-3-
x 2x
f ( x )
2x 4f ( 2 )
x 2
2x 4f ( 2 )
x 2
x 2
2
x 2 x 2 x 2
x 2
f ( 2 ) 2 " Given"
for lim f(x):
x 2 x 2x 4 0left limit : f( 2 ) f( 2 ) lim lim lim x 2 4
x 2 0 x 2
f ( 2 ) lim f(x) f ( x ) is discontinous at x 2
Note : we can redefine f(x) to make it continous by puttin
x 2
2
g f ( 2 ) 4 , so that we will
make f ( 2 ) lim f(x) 4
x 4
: x 2So , the Correct f(x) to be Continous is f(x) x 2
4 : x = 2
2
Example (7)2x 4
: x 2let f(x) Then Discuss the Continuity at x 2x 2
2 : x = 2
Answer
----------------------------------------------------------------------------------------- --------------Example (8)
3
2
x 8Redefine the following function f(x) to be Continous at x 2
2x x 6
Answer
Note :To redefine any function to be Continous , you have to find its limit and equalize it to
its value of f(x) at x = 2
2 2 23
2x 2 x 2 x 2
x 2
3
2
x 2 x 2x 4 x 2x 4 2 2 2 4x 8 0 12lim lim lim
2x x 6 0 2x 3 x 2 2x 3 2 2 3 7
12And f(x) is Continous at x 2 f ( 2 ) lim f ( x )
7
x 8x 2
2x x 6 So f(x)= 12
x 27
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Calculus 3rdsecondary Continuitystage (2)-3-
Example (9)
9 x 3Redefine the following function f (x) to be Continous at x 0
x
Answer
x 0 x 0 x 0
x 0
x 0
9 x 3 0 9 x 3 9 x 3 9 x 9lim lim lim
x 0 x 9 x 3 x 9 x 3
1 1 1lim
3 3 69 x 3
1And f(x) is Continous at x 0 f (0 ) lim f ( x )
6
9 x 3 x 0xSo f(x)=1
x 06
---------------------------------------------------------------------------------------------------------------------
Example (10)
4
x 3Redefine the following function f (x) to be Continous at x 3
x 81
Answer
2 2 2
2 2 2
2
2
x 3 x 3, x 3 , x 3
x 9 x 9 x 3 x 3 x 9f ( x ) f ( x )
x 3 x 3, x 3 , x 3
x 9 x 9 x 3 x 3 x 9
1, x 3
x 3 x 9
f ( x ) 1, x 3
x 3 x 9
2 2x 3 x 3
x 3
-1 -1 1 1f ( 3 ) lim And f ( 3 ) lim
108 108x 3 x 9 x 3 x 9
f ( 3 ) f ( 3 ) lim f ( x ) doesn't exist
So we cannot define f at x 3 to be continuous.
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0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Continuitystage (2)-3-
Example (11)
Tan 5 xRedefine the following function f (x) to be Continous at x 0
Sin 8x
Answer
x 0 x 0
Tan 5 xx 0
Tan 5 x 5 Sin 8 xlim f ( x ) lim So f(x)=
Sin 8 x 8 5x 0
8
---------------------------------------------------------------------------------------------------------------------
Example (12)2Cos x
Redefine the following function f (x) to be Continous at x1 Sin x 2
Answer
2 2
x x x x x2 2 2 2 2
x2
2
1 sin x 1 sin xcos x 0 1 sin xlim f ( x ) lim So lim lim lim 1 sin x 2
1 sin x 0 1 sin x 1 sin x
f(x) is Continous at x f ( ) lim f ( x ) 22 2
cos xx
1 sin x 2So f(x)=
2 x2
---------------------------------------------------------------------------------------------------------------------
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Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Continuitystage (2)-3-
2
x 2
2
x 2
x 2
x 3
At x 2 : f ( 2 ) 2 2 2 2 2
for lim f(x):
left limit : f( 2 ) lim x 2x 2 2
Right limit : f( 2 ) lim - x 3 1
f( 2 ) f( 2 ) limit does't exist at x 2 f ( x ) is discontinous at x 2
At x 3 : f ( 3 ) 3 3 0
for lim f(x
x 3
x 3
x 3 x 3
):
left limit : f( 3 ) lim - x 3 0
Right limit : f( 3 ) lim x 3 0
f( 3 ) f( 3 ) lim f(x) 3 f ( 3 ) lim f(x)
So f ( x ) is Continous at x 3
x 2x
f ( x ) 2x 2x 2 - x 3
x 3
x 32 0
Example (13)2x 2x 2 : x 2
let f(x) Then Discuss the Continuity at x 2 and x 3x 3 : x > 2
Answer
2
x 3 , x 3x 3
- x 3 , x 3
x 2x 2 , x 2x 3 , x 3
So for x > 2 : x 3 Then f(x) - x 3 , 2 x 3- x 3 , 2 x 3
x 3 , x 3
----------------------------------------------------------------------------------------- --------------
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0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Continuitystage (2)-4-
x 1
x 1
22
x 1 x 1 x 1
x 1
At x 1: f (1) 0 " given"
for lim f(x):
left limit : f( 1 ) lim 0 0
2 x 1 2 x 1 x 12x 2 0Right limit : f( 1 ) lim lim lim
x 1 0 x 1 x 1
lim 2 x 1 4
f( 1 ) f( 1 ) limit does't exist at x 1 f ( x ) is di
2
x 2
2
x 2
x 2
x 2 x 2
scontinous at x 2
And we can't redefine f(x)
2 2 2At x 2 : f ( 2 ) 6
2 1
for lim f(x)
2 x 2left limit : f( 2 ) lim 6
x 1
Right limit : f( 2 ) lim 10 2x 10 2(2 ) 6
f( 2 ) f( 2 ) lim f(x) 6 f ( 2 ) lim f(x)
So
f ( x ) is Continous at x 3
x 0x
f ( x ) Zero
22x 2
x 1
10 2x0
x 1
0
x 2
6
x 3
0
Example (14)
2
0 : 0 x 1
2x 2let f(x) : 1 x 2x 1
10 2x : 2 x 3
Then Discuss the Continuity at x 1 and x 2 , And if f(x) is discontinous , redefine f(x) if
possible
Answer
----------------------------------------------------------------------------------------- --------------
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Calculus 3rdsecondary Continuitystage (2)--
Example (15)
a x + 3 : x 1find the value of "a" to make f ( x ) Continous at x 1
3 a x : x < 1
Answer
x 1
x 1
f (1) a 3 , f ( 1 ) lim ( 3 a x ) 3 a
f (1 ) lim ( ax 3 ) a 3
f ( x ) is Continous at x 1 f ( 1 ) f (1 )
3 a a 3 a 0
----------------------------------------------------------------------------- --------------------------
Example (16)
find the value of a in the following function to make f(x) Continous
2x a 1 x a, x 1
f ( x ) at x 1x 1
2 , x = 1
Answer
x 1
2
x 1 x 1 x 1
x 1
f (1) 2 " Given"
for lim f(x):
x a 1 x a x a x 10f( 1 ) f( 1 ) lim lim lim x a 1 a
x 1 0 x 1
f ( x ) is Continous at x 1 f (1) lim f(x) 1 a 2 a 1
----------------------------------------------------------------------------------------- --------------
7/29/2019 Continuity stage (2)
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Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Continuitystage (2)--
x 0
x 0
x 0 x 0
x 0
f (0 ) a " Given"
for lim f(x):
5x Tan3x 0f( 0 ) f( 0 ) lim Divide both numerator and denominator by x
Sin4x 05x Tan3x Tan3x
55 3x x xlim lim 2
Sin4x Sin4x 4
x x
f ( x ) is Continous at x 0 f (0 ) lim f(x) a 2
Example (17)
5x Tan 3x: x 0
Sin4xIf f(x) is Continous at x 0 , Then find the value of "a"
a : x 0
Answer
----------------------------------------------------------------------------------------- --------------Example (18)
2
Find a and b, such that
x ax 2 : x 2
f ( x ) 4 : x 2 becomes continuous at x 2
5a bx : x 2
Answer
x 2
2
x 2
x 2
x 2
f ( 2 ) 4 " Given"
for lim f(x):
f( 2 ) lim x a x 2 4 2a 2 2a 2
f( 2 ) lim 5a bx 5a 2b
f ( x ) is Continous at x 2 f ( 2 ) lim f(x)
f ( 2 ) f ( 2 ) f ( 2 ) f ( 2 ) f ( 2 ) 4
2a 2 4 2a 2 a 1 Then substitu
te in the second equation
1And 5a 2b 4 5 2b 4 b -
2
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Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Continuitystage (2)-3-
x 0x
f ( x ) Sin2x
f 22x
x
f 2Tanx
x 0
x 0
Sin 2xRemember that : Sin 2x 2 Sin x Cos x Sin x Cos x
2
Sin 2x: x 0
2x
So f(x) a : x 0
x: x 0
Tan x
f (0 ) a " Given"
for lim f(x):
Sin 2x 0left limit : f( 0 ) lim
2x
x 0
x 0 x 0
x 0
x 0
Sin 2x 2lim 1
0 2x 2
x 0 xRight limit : f( 0 ) lim lim 1
Tan x 0 Tan x
f( 0 ) f( 0 ) lim f(x) 1
f(x) is Continous at x 0 f (0 ) lim f(x) a 1
a
Example (19)
If the function f ( x ) is continuous at x 0 , Then find a , where :
1
sin xcos x : x < 0x
f(x) a : x = 0
xcos x: x > 0
sin x
Answer
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0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Continuitystage (2)--
2
x a
x a
22 3
x a
3
x a
3
a x x a a x 4 x aa x f ( x )
- a x x a 2ax 3b x a
f ( a ) a a 4 4
for lim f(x):
f( a ) lim a x 4 4
f( a ) lim 2ax 3b 2a a 3b 2a 3b
f ( x ) is Continous at x 0 f ( a ) lim f(x) 2a 3b 4 (1)
And f ( a 4 ) b
3 3
3 3 3
a a 4 4 b b 8 b 2
Substitute in (1) to get a , 2a 3( 2 ) 4 2a 2 a 1 a 1
Example (20)
3
2
a x 4 : x aIf f(x) is Continous at x a , and f ( a 4 ) b
2ax 3b : x a
Then find the value of a and b
Answer
----------------------------------------------------------------------------------------- --------------
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0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Continuitystage (2)--
x 1
x 1
x 1
x 1
x 2
At x 1: f (1) a b
for lim f(x):
left limit : f( 1 ) lim x 2 1
Right limit : f( 1 ) lim ax b a b
f ( x ) is Continous at x 1 f (1) lim f(x) a b -1 (1)
At x 2 : f ( 2 ) 2a b
for lim f(x):
left limit : f( 2 ) l
x 2
x 2
x 2
im ax b 2a b
Right limit : f( 2 ) lim x 2 2
f ( x ) is Continous at x 2 f ( 2 ) lim f(x) 2a b 2 ( 2 )
from (1) and (2) by using subtraction : a 3 and by substituting b -4
x 1x
f ( x ) x 2 ax b
x 2
x 2
Example (21)
let f ( x ) be a continuous function at x 1 and at x 2 , then find a and b ,where :
x 2 : x 1
f ( x ) a x b : 1 x 2
x 2 : x 2
Answer
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Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Continuitystage (2)--
Continui ty of a function Over an interval
Any function is said to be Continuous over an intervalif the function is Continuous at every
point in the interval .
For example: f(x) = 7 is a constant function , its domain is R and the limit for eachpoint in Ris 7 , So the function is Continuous on R .
SO , Any Constant function is Continuous over R
Another example: f(x) = 4 23x 8x 7x 3 is a polynomial function , its domain is R and the
limit for any point in R equals the value of the function at the same point .let me Convince you
4 2
4 2 4 2
x a x a
x a
let a R where a is any point you have
So f ( a ) 3a 8a 7a 3
for lim f(x): lim 3x 8x 7x 3 3a 8a 7a 3
So as you see , at x a f ( a ) lim f ( x ) and that's true for each a R
f(x) is Continuous on R
SO , Any Polynomial function is Continuous over R
Another example: f(x) =1
x 1is Continuous on R {1}
But what if we ask you about the Continuity of f(x) =1
x 1on the interval ] 0 , 2 [
Then the answer would be f(x) is discontinuous over ]0 ,2 [ because f(x) is not defined at
x=1 which belongs to ]0 , 2 [ .
Remember that: each point belongs to the interval must be defined in order to make f(x) a
Continuous function over this interval .
SO , Any rational function is Continuous on R
except the set of zeroes of its denominator
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Calculus 3rdsecondary Continuitystage (2)--
The above can be summerized in the following :
2 n
0 1 2 n
2 n
0 1 2 n
2 m
0 1 2 m
( 1) The polynomial function f ( x ) a a x a x .... a x , is continous on R
or any subset of R .
p( x ) a a x a x .... a x( 2 ) The rational function f ( x )
Q( x ) b b x b x .... b x
is continuous on
R or any subset of R except at the zero s of Q( x )
( 3 ) The Trigonometric function.
a ) f ( x ) Sin x is continunous on R or any subset or R
b ) f ( x ) Cos x is continunous on R or any subset or R
c ) f ( x
) Tan x is continunous on R or any subset or R . exceptfor
2n 1 5 3The point s x : n Z " , , , ,........"
2 2 2 2 2
----------------------------------------------------------------------------------------- --------------
Simple examplesDiscuss the Continuity of f(x) on R of each of the following :
(1) f(x)=5
Answer
f(x) is Constant function , then it is Continuous on R
----------------------------------------------------------------------------------------- --------------
(2) f(x) = 23x 4x 3 Answer
f(x) is a polynomial function , then it is Continuous on R
----------------------------------------------------------------------------------------- --------------
(3) f(x) =3
x 2
Answer
The domain of f(x) is R {2} , Then f(x) is Continuous on R {2}
----------------------------------------------------------------------------------------- --------------
(4) f(x) =2
2
2x 3
x 9
Answer
The domain of f(x) is : 2 2x 9 0 x 9 x 3
Then f(x) is Continuous on R { 3 }
----------------------------------------------------------------------------------------- --------------
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Calculus 3rdsecondary Continuitystage (2)--
2
2 2
The domain of x 1 :
x 1 0 x 1
Continuous on R
The domain of 2x 3 :
3
2x 3 0 2x 3 x 2
3Continuous on R
2
3R
2
2
2
cos x sin x is continuous on R and x 3 is continuous on R .
f ( x ) is continuous on R except at the zeros of denominator
f ( x ) is continuous on R {-3}
3
x 1f ( x )
cos x
3
nf ( x ) is continuous on R x : x , n Z
2
(5) f(x) =2
x 3
x 4x 3
Answer
The domain of f(x) is 2x 4x 3 0 x 3 x 1 0 x 3 and x 1 Then f(x) is Continuous on R {3 , 1}
----------------------------------------------------------------------------------------- --------------
(6) f(x) =2x 1
2x 3
Answer
Then f(x) is Continuous on
----------------------------------------------------------------------------------------- --------------
(7)2
2xf ( x )
x 5x 8
Answer
2b 4ac for the denominator 25 4 1 8 25 32 -7 0Therefore there are no real zeroes for the denominator f is continuous on R .
----------------------------------------------------------------------------------------- --------------
1( 8 ) f ( x ) x 2 sin x Answer
1 2Each of the two functions g ( x ) x 2 "Polynomial fn" and g ( x ) sin x "Sin fn"
is continuous on R .
----------------------------------------------------------------------------------------- --------------
2
cos x sin x( 9 ) f ( x )
x 3
Answer
----------------------------------------------------------------------------------------- --------------
(10)
Answer
which makes Cos x = 0
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0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Continuitystage (2)--
The domain of x 3 :
x 3 0 x 3 Domain is 3,
x 3x
f ( x ) f ( 3 ) x 3
Examples(1) Discuss the Continuity of : f ( x ) x 3 over its domain .
Answer
At x = 3 : f(3)= 3 3 0
x 3for lim f(x):
x 3 x 3lim x 3= lim x 3 0
So as f(3) =x 3lim f ( x ) Then f(x) is Continuous at x 3
On the interval 3, :
x a x a
x a
let a R
So f ( a ) a 3
for lim f(x): lim a 3 a 3
f ( a ) lim f ( x ) f(x) is Continuous on R
----------------------------------------------------------------------------------------- --------------
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0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Continuitystage (2)-4-
x -3
x -3
x 3
2
x 2
x 2
At x -3 : f ( -3 ) 3 -3 2 -7
for lim f(x):
Right limit : f( -3 ) lim 3x 2 3 -3 2 -7
f ( 3 ) lim f(x) f ( x ) is Continous at x -3
At x 2 : f ( 2 ) 2 4 8
for lim f(x):
left limit : f( 2 ) lim 3x 2 8
Right limit
2
x 2
x 2
2
x 5
22
x 5
x 5
: f( 2 ) lim x 4 8
f ( x ) is Continous at x 2 f ( 2 ) lim f(x)
At x 5 : f ( 5 ) 5 4 29
for lim f(x):
left limit : f( 5 ) lim x 4 5 4 29
f ( 5 ) lim f(x) f ( x ) is Continous at x 5
x -3x
f ( x ) 3x 2
x 5
2x 4
x 2
7 8 29
2) Discuss the continuity of the function :2
3x 2 : -3 x 2f ( x )
x 4 : 2 x 5
Answer
For intervals :
(1) for all x -3,2 :
f ( x ) 3x 2 is a polynomial f ( x ) is continuous on - 3,2
for all x 2 , 5 : 2f ( x ) x +4 is a polynomial f ( x ) is continuous on x 2 , 5
(2) For points :
From (1) and ( 2 ) , we see that f ( x ) is continuous on -3 , 5
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Calculus 3rdsecondary Continuitystage (2)--
x 0x
f ( x ) Sin x Cos x
x
22Cos x 12 1
x 0
x 0
x 0
2
x
x
At x 0 : f (0 ) Sin 0 Cos 0 1
for lim f(x):
Right limit : f( 0 ) lim Sin x Cos x Sin 0 Cos 0 1
f (0 ) lim f(x) f ( x ) is Continous at x 0
At x : f ( ) 2Cos 1 1
for lim f(x):
left limit : f( ) lim Sin x Cos
2
x
x
x 1
Right limit : f( ) lim 2Cos x 1 1
f ( ) lim f(x) f ( x ) is Continuous at x
3) Discuss the continuity of the function :2
sin x cos x : 0 xf ( x )
2cos x 1 : x
Answer
For intervals :
(1) for all x 0 , : f ( x ) Sinx Cosx is continuous "Theorem"
for all x , 2f ( x ) 2Cos x 1 is continuous "Theorem"
(2) For points :
From (1) and ( 2 ) , we see that f ( x ) is continuous on 0 , ----------------------------------------------------------------------------------------- --------------
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Calculus 3rdsecondary Continuitystage (2)--
1x
2
1x
2
1x
2
1x
2
1 1 7 1At x : f ( ) 3
2 2 2 2
for lim f(x):
1 7 7 1left limit: f( ) lim x 3
2 2 2 2
1 3Right limit : f( ) lim 1
2 2x 4
1 1f( ) f( ) lim f(x) doesn' t exist
2 2
1f ( x ) is discontinous at x2
From (1) and ( 2 ) , we
1see that f ( x ) is continuous on R ,2
2
x
f ( x )7
x2
3
2x 4
1x
2
32x 4 0 2x 4 x 2
2x 4
3So the domain of is R- 22x 4
4) Discuss the continuity of the function on R where :
7 1x x
2 2f ( x )
3 1
x2x 4 2
Answer
For intervals :
(1)
1for all x - , :2
7 1f ( x ) x is a polynomial f ( x ) is continuous on - ,
2 2
1for all x , :
2
Domain of
1
f ( x ) is continuous on , 22
(2) For points :
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Calculus 3rdsecondary Continuitystage (2)-3-
x 0x
f ( x ) 23x 2
9 2x
x 5
5
x 1 x 2 x 3
2
x 0
2
x 0
x 0
x 1
2
x 1
x 1
At x 0 : f (0 ) 3 0 2 2
for lim f(x):
Right limit : f( 0 ) lim 3x 2 2
f (0 ) lim f(x) f ( x ) is Continous at x 0
At x 1: f (1) 5
for lim f(x):
left limit : f(1 ) lim 3x 2 5
Right limit : f( 1 ) lim 5 5
f ( x ) is Continous at x 1
23x +2 : 0 x 1
let f(x) 5 : 1 x 2
9 2x : 2 x 3x 5
(5) Then Discuss the Continuity over its domain
Answer
For intervals :
(1)
for all x 0,1 :
2f ( x ) 3x 2 is a polynomial f ( x ) is continuous on 0 ,1
for all x 1 , 2 :
f ( x ) 5 is a Constant function f ( x ) is continuous on 1 , 2
for all x 2 , 3 :
9 2xf ( x ) has a domain : x 5 But 5 2 , 3
x 5
f ( x ) is continuous on 2 , 3
(2) For points :
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Calculus 3rdsecondary Continuitystage (2)--
x 2
x 2
x 2
x 2
x 3
9 2 2 5At x 2 : f ( 2 ) -
2 5 3
for lim f(x):
left limit : f( 2 ) lim 5 5
9 2x 5Right limit : f( 2 ) lim -
x 5 3
f( 2 ) f( 2 ) lim f(x) doesn' t exist
f ( x ) is discontinous at x 2
9 2 3 3At x 3 : f ( 3 )
3 5 2
for lim
x 3
x 3
f(x):
9 2x 3left limit: f( 3 ) lim
x 5 2
f ( 3 ) lim f(x) f ( x ) is Continous at x 3
From (1) and ( 2 ) , we see that f ( x ) is continuous on 0 , 3 2
----------------------------------------------------------------------------------------- --------------
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Calculus 3rdsecondary Continuitystage (2)--
x 0
x 0
x 0
x 3
x 3
x 3
At x 0 : f (0 ) 5 0 5
for lim f(x):
Right limit : f( 0 ) lim 5 x 5
f (0 ) lim f(x) f ( x ) is Continous at x 0
At x 3 : f ( 3 ) 3 1 2
for lim f(x):
left limit : f( 3 ) lim 5 x 2
Right limit : f( 3 ) lim x 1 2
x 3f ( 3 ) lim f(x) f ( x ) is Continous at x 3
At x 4 : f ( 4 ) Undefined
f ( x ) is discontinous at x 4From (1) and ( 2 ) , we see that f ( x ) is continuous on 0 , 4
x
f ( x ) 5 x
x 4
x 1
x 3x 0
Discuss the continuity of the function : f ( x ) x 3 2 on 0 , 4 (6)
Answer
x 3 : x 3 x 3 : 0 x 3
x 3 x 3x 3 : x 3 x 3 : 3 x 4
3 x 2 : 0 x 3Thus f ( x ) x 3 2
x 3 2 : 3 x 4
5 x : 0 x 3f ( x )
x 1
: 3 x 4
For intervals :
(1) for all x 0,3 :
f ( x ) 5 x is a polynomial f ( x ) is continuous on 0 ,3
for all x 3 , 4 :
f ( x ) x 1 is a polynomial f ( x ) is continuous on x 3 , 4
(2) For points :
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Calculus 3rdsecondary Continuitystage (2)--
x
f ( x ) 5 2x 11 2x
3x
2
Discuss the continuity of the function : f ( x ) 8 3 2x in R
3x2
3x
2
3x
2
3 3 3At x : f ( ) 5 2 8
2 2 2
for lim f(x):
3left limit : f( ) lim 5 2x 8
2
3Right limit : f( ) lim 11 2x 8
2
3f ( x ) is Continous at x
2
From (1) and ( 2 ) , we see that f ( x ) is continuous on R
(7)
Answer
3 33 2x : x 8 3 2x : x2 2
3 2x f ( x )3 3
3 2x : x 8 3 2x : x2 2
35 2x : x
2Thus f ( x ) f ( x )
311 2x : x
2
For intervals :
(1)3
for all x - , :2
3f ( x ) 5 2x is a polynomial f ( x ) is continuous on - ,
2
3for all x , :2
3f ( x ) 11 2x is a polynomial f ( x ) is continuous on ,
2
(2) For points :
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Calculus 3rdsecondary Continuitystage (2)--
x
f ( x ) -2x 3 7
x 5x -2
Discuss the continuity of the function : f ( x ) x 2 x 5 in R
x 5 : x 5x 5
x 5 : x 5
x 2 : x 2x 2
x 2 : x 2
2x 3
x 2
x 2
x 2
x 5
x 5
At x 2 : f ( 2 ) 7
for lim f(x):
left limit : f( 2 ) lim 2x 3 7
Right limit : f( 2 ) lim 7 7
f ( x ) is Continuous at x 2
At x 5 : f ( 5 ) 2 5 3 7
for lim f(x):
left limit : f( 5 ) lim 7 7
Right limit : f( 5
x 5) lim 2x 3 7
f ( x ) is Continuous at x 5
From (1) and ( 2 ) , we see that f ( x ) is continuous on R
(8)
Answer
Forx 5 : f(x)= (x 5) + (x+2)= 2 x 3Forx 2 : f(x)= -( x + 2) ( x 5 )= -2 x + 3
For 2 x 5 : f(x)= ( x + 2) ( x 5 )= 7
For intervals :
(1) for all x - , 2 :
f ( x ) -2x 3 is a polynomial f ( x ) is continuous on - , 2
for all x 2 , 5 :
f ( x ) 7 is Constant function So , it is continuous on 2 , 5
for all x 5 , : f ( x ) 2x 3 is a polynomial f ( x ) is continuous on 5 ,
(2) For points :
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Calculus 3rdsecondary Continuitystage (2)--
2
ax 7 : x -1
If f(x) x +b : -1 x 2
3x a : x 2
x
f ( x ) 3x 2 a x b
x 5x -2
2x 12
x
f ( x ) ax 7 2x b
x 2x -1
3x a
x 2
f ( x ) is continuous on R
f ( x ) is continuous at x -2 f ( 2 ) 3 2 2 -8
And f ( -2 ) f ( -2 ) f ( -2 )
lim ax b -2a b -8 (1)
f ( x ) is continuous on R
f ( x ) is continuous
2
x 5
at x 5 f ( 5 ) 5 12 13
f ( 5 ) f ( 5 ) f ( 5 )
lim ax b 13 5a b 13 ( 2 )
From (1) and ( 2 ) and by subtraction : we get 7a 21 a 3 and b -8 6 -2
2
x -1
f ( x ) is continuous on R
f ( x ) is continuous at x -1 f ( 1) 1 b 1 b
And f ( -1 ) f ( -1 ) f ( -1)
lim ax 7 1 b a 7 1 b a b 6 (1)
And f ( x ) is continuous on R
f ( x )
2
x 2
is continuous at x 2 f ( 2 ) 3 2 a 6 a
f ( 2 ) f ( 2 ) f ( 2 )
lim x b 6 a 4 b 6 a a b 2 ( 2 )
From ( 1) and ( 2 ) and by addition: we get 2a 4 a 2 and b 4
2
3x 2 : x -2
(9 ) If the function f ( x ) ax b : - 2 x 5 is Continuous on R , then find a and b
x - 12 : x 5
Answer
----------------------------------------------------------------------------------------- --------------
(10) is Continuous on R , then find a and b
Answer
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Calculus 3rdsecondary Continuitystage (2)--
x 0
x 0 x 0
x 0
x3
At x 0 : f (0 ) 1 Cos 0 2
for lim f(x):
sin2x 0 sin2xleft limit : f( 0 ) lim lim 2
x 0 x
Right limit : f( 0 ) lim 1 Cos x 2
f ( x ) is Continuous at x 03
At x : f ( ) 1 Cos3 3 3 2
for lim f(x):
left limit : f(3
x3
3) lim 1 Cos x
2
f ( x ) is Continuous at x 3
From (1) and ( 2 ) , we see that f ( x ) is continuous on - ,6 3
x
f ( x )sin2x
x
x3
1 Cos x
x 0x
6
(11) Discuss the Continuity of
sin2x -: x 0
x 6f ( x )
1 cos x : 0 x 3
Answer
For intervals :
(1)for all x - , 0 :6
sin2xf ( x ) is a Sin function f ( x ) is continuous on - , 0
x 6
for all x 0 , :3
f ( x ) 1 Cos x is Cos function So , it is continuous on 0 ,3
(2) For points :
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x
f ( x )
x sin2x
1sin x
2
x a
x x - x 0
x 0
x 0
f ( x ) is continuous on 0 ,
f ( x ) is continuous at x 0 f (0 ) 0 a a
And f (0 ) f (0 ) f (0 )
x sin2x 0lim " Divide both numerator and denominator by x"
1 0sin x
2
sin2x1 1 2xlim 6 a 6 1 1
sin x2 2
x
2
x 5find the value of a and b such that f ( x ) is continuous where f ( x ) on R
x ax 25
(12)
Answer
2 2
2
f ( x ) is continuous on R The denominator has no zeros
Discriminant " b 4ac" is negative a 4 25 0
a 100 a -10 ,10
----------------------------------------------------------------------------------------- --------------(13 ) find the value of a and b such that f ( x ) is continuous where:
x sin2x: - x 0
1sin xf ( x )2
x a : 0 x
Answer
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