RD Sharma - Mathematics (Volume 1)

Book: RD Sharma - Mathematics (Volume 1)

Chapter: 9. Continuity

Subject: Maths - Class 12th

Q. No. 10 of Exercise 9.1

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10

Discuss the continuity of the following functions at the indicated point(s).

at
x = a

In this problem we need to check continuity at x = a

Given function is


at x = a …… Equation 2


∴ we need to check LHL, RHL and value of function at x = a (for idea and meaning of continuity refer to Q10(i)


Clearly,


f(a) = 0 [from equation 2]


LHL = [∵ sin(–θ) = sinθ]


RHL = [from eqn 2]


Why


Since whatever is value of h, sin(1/h) is going to range from –1 to 1


As h→ 0 ,i.e. approximately 0


∴ 0*some finite quantity is equal to 0.


Clearly, LHL = RHL = f(a)


∴ f(x) is continuous at x = 0


Chapter Exercises

Exercise 9.1
Exercise 9.2
Very short answer
MCQ

More Exercise Questions

1

Test the continuity of the following function at the origin :
2

A function f(x) is defined as Show that f(x) is continuous at x = 3.
3

A function f(x) is defined as

Show that f(x) is continuous at x = 3.
4

If Find whether f(x) is continuous at x = 1.
5

If Find whether f(x) is continuous at x = 0.
6

If Find whether f is continuous at x = 0.
7

Let Show that f(x) is discontinuous at x = 0.
8

Show that is discontinuous at x = 0.
9

Show that is discontinuous at x = a.
10

Discuss the continuity of the following functions at the indicated point(s).

at x = 0
10

Discuss the continuity of the following functions at the indicated point(s).

at x = 0
10

Discuss the continuity of the following functions at the indicated point(s).

at
x = a
10

Discuss the continuity of the following functions at the indicated point(s).

at x = 0
10

Discuss the continuity of the following functions at the indicated point(s).

at x = 1
10

Discuss the continuity of the following functions at the indicated point(s).

at x = 1
10

Discuss the continuity of the following functions at the indicated point(s).

at x = 0
10

Discuss the continuity of the following functions at the indicated point(s).

at x = a
11

Show that is discontinuous at x = 1.
12

Show that is continuous at x = 0
13

Find the value of ‘a’ for which the function f defined by is continuous at x = 0.
14

Examine the continuity of the function at x = 0. Also sketch the graph of this function.
15

Discuss the continuity of the function at the point x = 0.
16

Discuss the continuity of the function at the point x = 1/2.
17

Discuss the continuity of at x = 0.
18

For what value of k is the function continuous at
x = 1?
19

Determine the value of the constant k so that the function is continuous at x = 1.
20

For what value of k is the function continuous at
x = 0?
21

Determine the value of the constant k so that the function is continuous at x = 2.
22

Determine the value of the constant k so that the function is continuous at x = 0.
23

Find the values of a so that the function is continuous at x = 2.
24

Prove that the function remains discontinuous at x = 0, regardless of the choice of k.
25

Find the value of k if f(x) is continuous at x = π/2, where

26

Determine the values of a, b, c for which the function is continuous at x = 0.
27

If is continuous at x = 0, find k.
28

If is continuous at x = 4, find a, b.
29

For what value of k is the function continuous at x = 0?
30

Let x ≠ 0. Find the value of f at x = 0 so that f becomes continuous at x = 0.
31

If is continuous at x = 2, find k.
32

If is continuous at x = 0, find k.
33

Extend the definition of the following by continuity at the point x = π.
34

If is continuous at x = 0, then find f(0).
35

Find the value of k for which is continuous at x = 0.
36

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point :

at x = 0
36

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point :

at x = 1
36

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point :

at x = 0
36

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point :

at x = 5
36

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point :

at x = 5
36

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point :

at x = 1
36

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point :

at x = 0
36

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point :

at x = 2.
37

Find the values of a and b so that the function f given by is continuous at x = 3 and x = 5
38

If Show that f is continuous at x = 1.
39

Discuss the continuity of the f(x) at the indicated points :

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39

Discuss the continuity of the f(x) at the indicated points :

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40

Prove that is discontinuous at

x = 0.
41

If then what should be the value of k so that f(x) is continuous at x = 0.
42

For what value of is the function continuous at x = 0? What about continuity at x = ± 1?
43

For what value of k is the following function continuous at x = 2?
44

Let . If f(x) is continuous at
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45

If the function f(x), defined below is continuous at x = 0, find the value of k:
46

Find the relationship between ‘a’ and ‘b’ so that the function ‘f’ defined by is continuous at x = 3.

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