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Calculus 1

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1
By Gökhan Bilhan
Calculus 1
(Week 5)-Continuity
Denition(Continuity) A function
properties holds:
f is continuous at the point x = c, if each of the following
1-) limx→c f (x) exists.
2-) f (c) exists.
3-) limx→c f (x) = f (c) (Means, we can put x = c at f (x).)
Remark. If one of the above properties fails to be true, then f is not continuous at x = c.
Denition A function is continuous on the open interval (a, b), if it is continuous at each point on
the interval.
Example
{
2, if
Given the function f (x) =
1, if
(a) Graph f .
x
x
is an integer
is not an integer
(b) limx→2 f (x)
(d) Is f continuous at x = 2? What are the discontinuity points of f ?
(c) f (2)
2
By Gökhan Bilhan
Example Discuss the continuity of the function f (x) = x + 2 at x = 2 and at any point.
Remark. All polynomial functions are continuous everywhere.
Example
{
1 + x, if
Find all discontinuity points of f (x) =
5 − x, if
x≤2
x>2
Example Use the denition of continuity to discuss the continuity of the function whose graph is
below.
3
By Gökhan Bilhan
Remark If we can draw a function without raising our hand, then the function is continuous there.
Exercises
{
1 + x, if
1. Find the points of discontinuity of f (x) =
5 − x, if


−x, if
2. Find the points of discontinuity of f (x) = 1,
if


x,
if
x<1
x≥1
x<0
x=0
x>0
By Gökhan Bilhan
3. Here is the question
{
−1, if x is an even integer
4. Given the function g(x) =
1,
if x is not an even integer
(a) Graph g
(b) limx→1 g(x) =?
(c) g(1)=?
(d) Is g cont. at x = 1?
(e) Where is g discontinuous?
4
By Gökhan Bilhan
5. Here is the question
6. Here is the question
5
6
By Gökhan Bilhan
Let's draw the graphs of y =
limx→0+
1
=
x
limx→0+
1
=
x2
1
1
and y = 2
x
x
limx→0−
limx→0−
Example How to nd limx→−1
In particular, limx→−1+
1
=
x
1
=
x2
limx→0
1
=
x
limx→0
1
=
x2
x−1
x+1
x−1
=?
x+1
limx→−1−
x−1
=?
x+1
7
By Gökhan Bilhan
Examples
limx→1
1
=?
x−1
limx→1
1
=?
(x − 1)2
limx→2
x2 + 20
=?
5(x − 2)2
Remark limx→∞
1
=
x
limx→−∞
1
=
x
8
By Gökhan Bilhan
Example limx→−∞
11x + 2
2x3 − 1
Example limx→∞ 2x3 − x2 − 7x + 3 =
Exercises
x2
=?
1. limx→−3
x+3
2. limx→−2
2x + 2
=?
(x + 2)2
9
By Gökhan Bilhan
x2 + 2x − 3
3. limx→∞ 2
=?
x − 4x + 3
√
4. limx→∞ x2 + 1 − x =?
5. limx→∞ x2 − x =
6. limx→∞
x2 + x
=
3−x
√
√
3
x− 5x
√ =
7. limx→−∞ √
3
x+ 5x
10
By Gökhan Bilhan
(Week 5)-Continuity(Exercises)
Exercises
√
3x + 4x2 + 7x − 1
√
1. limx→∞
=?
x + x2 − x + 4
A)5
B)3,5
2. What is
a,
C)2,5
B)0
3. What is
n,
x≤1
1<x
if
if
C)1
1
D) 2
3
E) 2
to say the following function is continuous everywhere.


mx + n,
f (x) = 5,

 2
x + m,
A)-2
E)0
to say the following function is continuous everywhere.
{
1 + x,
f (x) =
3 − ax2 ,
A)2
D)1,5
B)-1
if
if
if
C)1
1>x
x=1
x>1
D)2
E)4
4. At which of the following points is the function
f (x) =
A)2
|x2
1
− 4|
+ 2
x2 − 4
x −1
B)-2
C)1
D)0
E)-1
f
continuous?
11
By Gökhan Bilhan
5.
limx→2 (
A)
6.
1
4
− 2
)=
x−2 x −4
−1
8
B)
limx→4 ( √
A)4
−1
4
C)0
D)
1
4
E)
1
8
1
4
−
)=
x−2 x−4
B)3
C)2
D)
1
2
E)
1
4
√
7.
√
x + 3 − 3x + 1
√
limx→1 (
)=
x−1
A)−∞
B)−2
√
2+ x
√ =
2− x
8.
limx→∞
9.
limx→0− (
2
1
x5
)=
C)−1
D)0
E)4
12
By Gökhan Bilhan
10.
11.
limx→0 (
1
2
)=
x3
√
3x + 4x2 + 7x − 1
√
limx→−∞
=?
x + x2 − x + 4
Answers:
1-) C , 2-)C , 3-)C , 4-)D , 5-)D , 6-)E , 7-)D
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