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RD Sharma solutions for Class 12 Mathematics chapter 9 - Continuity

Chapter 9 : Continuity

Pages 16 - 21

Q 1 | Page 16

Test the continuity of the function on f(x) at the origin:

$f\left( x \right) = \begin{cases}\frac{x}{\left| x \right|}, & x \neq 0 \\ 1 , & x = 0\end{cases}$

Q 2 | Page 16

A function f(x) is defined as,

$f\left( x \right) = \begin{cases}\frac{x^2 - x - 6}{x - 3}; if & x \neq 3 \\ 5 ; if & x = 3\end{cases}$  Show that f(x) is continuous that x = 3.
Q 3 | Page 16

A function f(x) is defined as

$f\left( x \right) = \begin{cases}\frac{x^2 - 9}{x - 3}; if & x \neq 3 \\ 6 ; if & x = 3\end{cases}$

Show that f(x) is continuous at x = 3

Q 4 | Page 17

If $f\left( x \right) = \begin{cases}\frac{x^2 - 1}{x - 1}; for & x \neq 1 \\ 2 ; for & x = 1\end{cases}$ Find whether f(x) is continuous at x = 1.

Q 5 | Page 17

If $f\left( x \right) = \begin{cases}\frac{\sin 3x}{x}, when & x \neq 0 \\ 1 , when & x = 0\end{cases}$

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

Q 6 | Page 17

If $f\left( x \right) = \begin{cases}e^{1/x} , if & x \neq 0 \\ 1 , if & x = 0\end{cases}$ find whether f is continuous at x = 0.

Q 7 | Page 17

Let $f\left( x \right) = \begin{cases}\frac{1 - \cos x}{x^2}, when & x \neq 0 \\ 1 , when & x = 0\end{cases}$ Show that f(x) is discontinuous at x = 0.

Q 8 | Page 17

Show that

$f\left( x \right) = \begin{cases}\frac{x - \left| x \right|}{2}, when & x \neq 0 \\ 2 , when & x = 0\end{cases} is discontinuous at x = 0. Q 9 | Page 17 Show that \[f\left( x \right) = \begin{cases}\frac{\left| x - a \right|}{x - a}, when & x \neq a \\ 1 , when & x = a\end{cases}$ is discontinuous at x = a.
Q 10.1 | Page 17

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

(i) $f\left( x \right) = \begin{cases}\left| x \right| \cos\left( \frac{1}{x} \right), & x \neq 0 \\ 0 , & x = 0\end{cases}at x = 0$

Q 10.2 | Page 17

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

(ii) $f\left( x \right) = \left\{ \begin{array}{l}x^2 \sin\left( \frac{1}{x} \right), & x \neq 0 \\ 0 , & x = 0\end{array}at x = 0 \right.$

Q 10.3 | Page 17

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

$f\left( x \right) = \left\{ \begin{array}{l}(x - a)\sin\left( \frac{1}{x - a} \right), & x \neq a \\ 0 , & x = a\end{array}at x = a \right.$

Q 10.4 | Page 17

Discuss the continuity of the following functions at the indicated point(s): (iv) $f\left( x \right) = \left\{ \begin{array}{l}\frac{e^x - 1}{\log(1 + 2x)}, if & x \neq a \\ 7 , if & x = 0\end{array}at x = 0 \right.$

Q 10.5 | Page 17

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

$f\left( x \right) = \left\{ \begin{array}{l}\frac{1 - x^n}{1 - x}, & x \neq 1 \\ n - 1 , & x = 1\end{array}n \in N \right.at x = 1$
Q 10.6 | Page 17

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

$f\left( x \right) = \begin{cases}\frac{\left| x^2 - 1 \right|}{x - 1}, for & x \neq 1 \\ 2 , for & x = 1\end{cases}at x = 1$
Q 10.7 | Page 17

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

$f\left( x \right) = \left\{ \begin{array}{l}\frac{2\left| x \right| + x^2}{x}, & x \neq 0 \\ 0 , & x = 0\end{array}at x = 0 \right.$
Q 10.8 | Page 17

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

$f\left( x \right) = \binom{\left| x - a \right|\sin\left( \frac{1}{x - a} \right), for x \neq a}{0, for x = a}at x = a$
Q 11 | Page 18

Show that

$f\left( x \right) = \begin{cases}1 + x^2 , if & 0 \leq x \leq 1 \\ 2 - x , if & x > 1\end{cases}$

Q 12 | Page 18

Show that

$f\left( x \right) = \begin{cases}\frac{\sin 3x}{\tan 2x} , if x < 0 \\ \frac{3}{2} , if x = 0 \\ \frac{\log(1 + 3x)}{e^{2x} - 1} , if x > 0\end{cases}\text{is continuous at} x = 0$

Q 13 | Page 18

Find the value of 'a' for which the function f defined by

$f\left( x \right) = \begin{cases}a\sin\frac{\pi}{2}(x + 1), & x \leq 0 \\ \ frac{\ tanx - \ sinx}{x^3}, & x > 0\end{cases}$

Q 14 | Page 18

Examine the continuity of the function

$f\left( x \right) = \left\{ \begin{array}{l}3x - 2, & x \leq 0 \\ x + 1 , & x > 0\end{array}at x = 0 \right.$

Also sketch the graph of this function.

Q 15 | Page 18

Discuss the continuity of the function f(x) at the point x = 0, where $f\left( x \right) = \left\{ \begin{array}x, \\ 1, \\ - x,\end{array}\begin{array}x > 0 \\ x = 0 \\ x < 0\end{array} \right.$

Q 16 | Page 18

Discuss the continuity of the function f(x) at the point x = 1/2, where $f\left( x \right) = \left\{ \begin{array}x, \\ 1/2, \\ 1 - x,\end{array}\begin{array}0 \leq x < 1/2 \\ x = 1/2 \\ 1/2 < x \leq 1\end{array} \right.$

Q 17 | Page 18

Discuss the continuity of $f\left( x \right) = \begin{cases}2x - 1 & , x < 0 \\ 2x + 1 & , x \geq 0\end{cases} at x = 0$

Q 18 | Page 18

For what value of k is the following function continuous at x = 1? $f\left( x \right) = \begin{cases}\frac{x^2 - 1}{x - 1}, & x \neq 1 \\ k , & x = 1\end{cases}$

Q 19 | Page 18

Determine the value of the constant k so that the function

$f\left( x \right) = \left\{ \begin{array}{l}\frac{x^2 - 3x + 2}{x - 1}, if & x \neq 1 \\ k , if & x = 1\end{array}\text{is continuous at x} = 1 \right.$

Q 20 | Page 18

For what value of k is the function

$f\left( x \right) = \begin{cases}\frac{\sin 5x}{3x}, if & x \neq 0 \\ k , if & x = 0\end{cases}\text{is continuous at x} = 0?$

Q 21 | Page 18

Determine the value of the constant k so that the function

$f\left( x \right) = \begin{cases}k x^2 , if & x \leq 2 \\ 3 , if & x > 2\end{cases}\text{is continuous at x} = 2 .$

Q 22 | Page 19

Determine the value of the constant k so that the function

$f\left( x \right) = \begin{cases}\frac{\sin 2x}{5x}, if & x \neq 0 \\ k , if & x = 0\end{cases}\text{is continuous at x} = 0 .$

Q 23 | Page 19

Find the values of a so that the function

$f\left( x \right) = \begin{cases}ax + 5, if & x \leq 2 \\ x - 1 , if & x > 2\end{cases}\text{is continuous at x} = 2 .$
Q 24 | Page 19

Prove that the function

$f\left( x \right) = \begin{cases}\frac{x}{\left| x \right| + 2 x^2}, & x \neq 0 \\ k , & x = 0\end{cases}$  remains discontinuous at x = 0, regardless the choice of k.
Q 25 | Page 19

Find the value of k if f(x) is continuous at x = π/2, where $f\left( x \right) = \begin{cases}\frac{k \cos x}{\pi - 2x}, & x \neq \pi/2 \\ 3 , & x = \pi/2\end{cases}$

Q 26 | Page 19

Determine the values of abc for which the function

$f\left( x \right) = \begin{cases}\frac{\ sin (a + 1)x + \ sinx}{x}, & for x < 0 \\ c , & for x = 0 \\ \frac{\sqrt{x + b x^2} - \sqrt{x}}{b x^{3/2}} , & for x > 0\end{cases}$ is continuous at x = 0.

Q 27 | Page 19

If  $f\left( x \right) = \begin{cases}\frac{1 - \cos kx}{x \sin x}, & x \neq 0 \\ \frac{1}{2} , & x = 0\end{cases}\text{is continuous at x} = 0, find k .$

Q 28 | Page 19

If $f\left( x \right) = \begin{cases}\frac{x - 4}{\left| x - 4 \right|} + a, if & x < 4 \\ a + b , if & x = 4 \\ \frac{x - 4}{\left| x - 4 \right|} + b, if & x > 4\end{cases}$  is continuous at x = 4, find ab.

Q 29 | Page 19

For what value of k is the function

$f\left( x \right) = \begin{cases}\frac{\sin 2x}{x}, & x \neq 0 \\ k , & x = 0\end{cases}$  continuous at x = 0?

Q 30 | Page 19

Let  $f\left( x \right) = \frac{\log\left( 1 + \frac{x}{a} \right) - \log\left( 1 - \frac{x}{b} \right)}{x}$ x ≠ 0. Find the value of f at x = 0 so that f becomes continuous at x = 0.

Q 31 | Page 19

If   $f\left( x \right) = \begin{cases}\frac{2^{x + 2} - 16}{4^x - 16}, if & x \neq 2 \\ k , if & x = 2\end{cases}$  is continuous at x = 2, find k.

Q 32 | Page 20

If  $f\left( x \right) = \begin{cases}\frac{\cos^2 x - \sin^2 x - 1}{\sqrt{x^2 + 1} - 1}, & x \neq 0 \\ k , & x = 0\end{cases}$   is continuous at x = 0, find k.

Q 33 | Page 20

Extend the definition of the following by continuity

$f\left( x \right) = \frac{1 - \cos7 (x - \pi)}{5 (x - \pi )^2}$
Q 34 | Page 20

Given:

$f\left( x \right) = \frac{2x + 3\ sinx}{3x + 2\ sinx}, x \neq 0$ If f(x) is continuous at x = 0, then
Q 35 | Page 20

Find the value of k for which $f\left( x \right) = \begin{cases}\frac{1 - \cos 4x}{8 x^2}, when & x \neq 0 \\ k , when & x = 0\end{cases}$ is continuous at x = 0;

Q 36.1 | Page 20

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; (i) $f\left( x \right) = \begin{cases}\frac{1 - \cos 2kx}{x^2}, if & x \neq 0 \\ 8 , if & x = 0\end{cases}$ at x = 0

Q 36.2 | Page 20

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; $f\left( x \right) = \begin{cases}(x - 1)\tan\frac{\ pix}{2}, if & x \neq 1 \\ k , if & x = 1\end{cases}$ at x = 1at x = 1

Q 36.3 | Page 20

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

$f\left( x \right) = \begin{cases}k( x^2 - 2x), if & x < 0 \\ \cos x, if & x \geq 0\end{cases}$ at x = 0
Q 36.4 | Page 20

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

$f\left( x \right) = \begin{cases}kx + 1, if & x \leq \pi \\ \cos x, if & x > \pi\end{cases}$ at x = 0
Q 36.5 | Page 20

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point;  $f\left( x \right) = \begin{cases}kx + 1, if & x \leq 5 \\ 3x - 5, if & x > 5\end{cases}$ at x = 5

Q 36.6 | Page 20

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point;  $f\left( x \right) = \begin{cases}\frac{x^2 - 25}{x - 5}, & x \neq 5 \\ k , & x = 5\end{cases}$at x = 5

Q 36.7 | Page 20

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point;  $f\left( x \right) = \begin{cases}k x^2 , & x \geq 1 \\ 4 , & x < 1\end{cases}$at x = 1

Q 36.8 | Page 20

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; $f\left( x \right) = \begin{cases}k( x^2 + 2), if & x \leq 0 \\ 3x + 1 , if & x > 0\end{cases}$

Q 36.9 | Page 20

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point; $f\left( x \right) = \binom{\frac{x^3 + x^2 - 16x + 20}{\left( x - 2 \right)^2}, x \neq 2}{k, x = 2}$

Q 37 | Page 20

Find the values of a and b so that the function f given by $f\left( x \right) = \begin{cases}1 , & if x \leq 3 \\ ax + b , & if 3 < x < 5 \\ 7 , & if x \geq 5\end{cases}$ is continuous at x = 3 and x = 5.

Q 38 | Page 21

If $f\left( x \right) = \begin{cases}\frac{x^2}{2}, & if 0 \leq x \leq 1 \\ 2 x^2 - 3x + \frac{3}{2}, & if 1 < x \leq 2\end{cases}$. Show that f is continuous at x = 1.

Q 39 | Page 21

Discuss the continuity of the f(x) at the indicated points:  f(x) = | x − 1 | + | x + 1 | at x = −1, 1.

Q 39.1 | Page 21

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

(i) f(x) = | x | + | x − 1 | at x = 0, 1.

Q 40 | Page 21

Prove that  $f\left( x \right) = \begin{cases}\frac{x - \left| x \right|}{x}, & x \neq 0 \\ 2 , & x = 0\end{cases}$ is discontinuous at x = 0

Q 41 | Page 21

If  $f\left( x \right) = \begin{cases}2 x^2 + k, & if x \geq 0 \\ - 2 x^2 + k, & if x < 0\end{cases}$  then what should be the value of k so that f(x) is continuous at x = 0.

Q 42 | Page 21

For what value of λ is the function
$f\left( x \right) = \begin{cases}\lambda( x^2 - 2x), & if x \leq 0 \\ 4x + 1 , & if x > 0\end{cases}$continuous at x = 0? What about continuity at x = ± 1?

Q 43 | Page 21

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

$f\left( x \right) = \begin{cases}2x + 1 ; & if x < 2 \\ k ; & x = 2 \\ 3x - 1 ; & x > 2\end{cases}$
Q 44 | Page 21

Let$f\left( x \right) = \left\{ \begin{array}\frac{1 - \sin^3 x}{3 \cos^2 x} , & if x < \frac{\pi}{2} \\ a , & if x = \frac{\pi}{2} \\ \frac{b(1 - \sin x)}{(\pi - 2x )^2}, & if x > \frac{\pi}{2}\end{array} . \right.$If f(x) is continuous at x =

$\frac{\pi}{2}$

Q 45 | Page 21

If the functions f(x), defined below is continuous at x = 0, find the value of k. $f\left( x \right) = \begin{cases}\frac{1 - \cos 2x}{2 x^2}, & x < 0 \\ k , & x = 0 \\ \frac{x}{\left| x \right|} , & x > 0\end{cases}$

Q 46 | Page 21

Find the relationship between 'a' and 'b' so that the function 'f' defined by

$f\left( x \right) = \begin{cases}ax + 1, & if x \leq 3 \\ bx + 3, & if x > 3\end{cases}$is continuous at x = 3.

Pages 21 - 37

Q 1 | Page 34

Prove that the function

$f\left( x \right) = \begin{cases}\frac{\sin x}{x}, & x < 0 \\ x + 1, & x \geq 0\end{cases}$  is everywhere continuous.

Q 2 | Page 34

Discuss the continuity of the function

$f\left( x \right) = \left\{ \begin{array}{l}\frac{x}{\left| x \right|}, & x \neq 0 \\ 0 , & x = 0\end{array} . \right.$
Q 3.01 | Page 21

Find the points of discontinuity, if any, of the following functions:

$f\left( x \right) = \begin{cases}x^3 - x^2 + 2x - 2, & if x \neq 1 \\ 4 , & if x = 1\end{cases}$

Q 3.02 | Page 34

Find the points of discontinuity, if any, of the following functions:

$f\left( x \right) = \begin{cases}\frac{x^4 - 16}{x - 2}, & if x \neq 2 \\ 16 , & if x = 2\end{cases}$
Q 3.03 | Page 34

Find the points of discontinuity, if any, of the following functions:

$f\left( x \right) = \begin{cases}\frac{\sin x}{x}, & if x < 0 \\ 2x + 3, & x \geq 0\end{cases}$
Q 3.04 | Page 34

Find the points of discontinuity, if any, of the following functions:

$f\left( x \right) = \begin{cases}\frac{\sin 3x}{x}, & if x \neq 0 \\ 4 , & if x = 0\end{cases}$

Q 3.05 | Page 34

Find the points of discontinuity, if any, of the following functions:

$f\left( x \right) = \begin{cases}\frac{\sin x}{x} + \cos x, & if x \neq 0 \\ 5 , & if x = 0\end{cases}$
Q 3.06 | Page 34

Find the points of discontinuity, if any, of the following functions:

$f\left( x \right) = \begin{cases}\frac{x^4 + x^3 + 2 x^2}{\tan^{- 1} x}, & if x \neq 0 \\ 10 , & if x = 0\end{cases}$
Q 3.07 | Page 34

Find the points of discontinuity, if any, of the following functions:

$f\left( x \right) = \begin{cases}\frac{e^x - 1}{\log_e (1 + 2x)}, & if x \neq 0 \\ 7 , & if x = 0\end{cases}$
Q 3.08 | Page 34

Find the points of discontinuity, if any, of the following functions:

$f\left( x \right) = \begin{cases}\left| x - 3 \right|, & if x \geq 1 \\ \frac{x^2}{4} - \frac{3x}{2} + \frac{13}{4}, & if x < 1\end{cases}$
Q 3.09 | Page 34

Find the points of discontinuity, if any, of the following functions:

$f\left( x \right) = \begin{cases}\left| x \right| + 3 , & if x \leq - 3 \\ - 2x , & if - 3 < x < 3 \\ 6x + 2 , & if x > 3\end{cases}$
Q 3.1 | Page 34

Find the points of discontinuity, if any, of the following functions:

$f\left( x \right) = \begin{cases}x^{10} - 1, & if x \leq 1 \\ x^2 , & if x > 1\end{cases}$

Q 3.11 | Page 34

Find the points of discontinuity, if any, of the following functions:

$f\left( x \right) = \begin{cases}2x , & if & x < 0 \\ 0 , & if & 0 \leq x \leq 1 \\ 4x , & if & x > 1\end{cases}$
Q 3.12 | Page 34

Find the points of discontinuity, if any, of the following functions:

$f\left( x \right) = \begin{cases}- 2 , & if & x \leq - 1 \\ 2x , & if & - 1 < x < 1 \\ 2 , & if & x \geq 1\end{cases}$
Q 4.1 | Page 35

In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous:

$f\left( x \right) = \begin{cases}\frac{\sin 2x}{5x}, & if x \neq 0 \\ 3k , & if x = 0\end{cases}$
Q 4.2 | Page 35

In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous:

$f\left( x \right) = \begin{cases}kx + 5, & if x \leq 2 \\ x - 1, & if x > 2\end{cases}$
Q 4.3 | Page 35

In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous:

$f\left( x \right) = \begin{cases}k( x^2 + 3x), & if x < 0 \\ \cos 2x , & if x \geq 0\end{cases}$
Q 4.4 | Page 35

In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous:

$f\left( x \right) = \begin{cases}2 , & if x \leq 3 \\ ax + b, & if 3 < x < 5 \\ 9 , & if x \geq 5\end{cases}$
Q 4.5 | Page 35

In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous: $f\left( x \right) = \begin{cases}4 , & if x \leq - 1 \\ a x^2 + b, & if - 1 < x < 0 \\ \cos x, & if x \geq 0\end{cases}$

Q 4.6 | Page 35

In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous: $f\left( x \right) = \begin{cases}\frac{\sqrt{1 + px} - \sqrt{1 - px}}{x}, & if - 1 \leq x < 0 \\ \frac{2x + 1}{x - 2} , & if 0 \leq x \leq 1\end{cases}$

Q 4.7 | Page 35

In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous: $f\left( x \right) = \begin{cases}5 , & if & x \leq 2 \\ ax + b, & if & 2 < x < 10 \\ 21 , & if & x \geq 10\end{cases}$

Q 4.8 | Page 35

In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous:

$f\left( x \right) = \begin{cases}\frac{k \cos x}{\pi - 2x} , & x < \frac{\pi}{2} \\ 3 , & x = \frac{\pi}{2} \\ \frac{3 \tan 2x}{2x - \pi}, & x > \frac{\pi}{2}\end{cases}$
Q 5 | Page 36

The function

$f\left( x \right) = \begin{cases}x^2 /a , & if 0 \leq x < 1 \\ a , & if 1 \leq x < \sqrt{2} \\ \frac{2 b^2 - 4b}{x^2}, & if \sqrt{2} \leq x < \infty\end{cases}$

is continuous on (0, ∞), then find the most suitable values of a and b.

Q 7 | Page 36

The function f(x) is defined as follows:

$f\left( x \right) = \begin{cases}x^2 + ax + b , & 0 \leq x < 2 \\ 3x + 2 , & 2 \leq x \leq 4 \\ 2ax + 5b , & 4 < x \leq 8\end{cases}$

If f is continuous on [0, 8], find the values of a and b.

Q 7 | Page 36

Find the values of a and b so that the function f(x) defined by $f\left( x \right) = \begin{cases}x + a\sqrt{2}\sin x , & if 0 \leq x < \pi/4 \\ 2x \cot x + b , & if \pi/4 \leq x < \pi/2 \\ a \cos 2x - b \sin x, & if \pi/2 \leq x \leq \pi\end{cases}$becomes continuous on [0, π].

Q 8 | Page 36

If $f\left( x \right) = \frac{\tan\left( \frac{\pi}{4} - x \right)}{\cot 2x}$

for x ≠ π/4, find the value which can be assigned to f(x) at x = π/4 so that the function f(x) becomes continuous every where in [0, π/2].

Q 9 | Page 36

Discuss the continuity of the function

$f\left( x \right) = \begin{cases}2x - 1 , & if x < 2 \\ \frac{3x}{2} , & if x \geq 2\end{cases}$
Q 10 | Page 36

Discuss the continuity of f(x) = sin | x |.

Q 11 | Page 37
Prove that
$f\left( x \right) = \begin{cases}\frac{\sin x}{x} , & x < 0 \\ x + 1 , & x \geq 0\end{cases}$is everywhere continuous.

Q 12 | Page 37

Show that the function g (x) = x − [x] is discontinuous at all integral points. Here [x] denotes the greatest integer function.

Q 13 | Page 37

Discuss the continuity of the following functions:
(i) f(x) = sin x + cos x
(ii) f(x) = sin x − cos x
(iii) f(x) = sin x cos x

Q 14 | Page 37

Show that f (x) = cos x2 is a continuous function.

Q 15 | Page 37

Show that f (x) = | cos x | is a continuous function.

Q 16 | Page 37

Find all the points of discontinuity of f defined by f (x) = | x |− | x + 1 |.

Q 17 | Page 37

Determine if $f\left( x \right) = \begin{cases}x^2 \sin\frac{1}{x} , & x \neq 0 \\ 0 , & x = 0\end{cases}$ is a continuous function?

Q 18 | Page 37
Given the function
$f\left( x \right) = \frac{1}{x + 2}$ . Find the points of discontinuity of the function f(f(x)).
Q 19 | Page 37

Find all point of discontinuity of the function

$f\left( t \right) = \frac{1}{t^2 + t - 2}, where t = \frac{1}{x - 1}$

Pages 41 - 42

Q 1 | Page 41

Define continuity of a function at a point.

Q 2 | Page 41

What happens to a function f (x) at x = a, if

$\lim_{x \to a}$ f (x) = f (a)?
Q 3 | Page 41

Find f (0), so that

$f\left( x \right) = \frac{x}{1 - \sqrt{1 - x}}$  becomes continuous at x = 0.

Q 4 | Page 41

If $f\left( x \right) = \begin{cases}\frac{x}{\sin 3x}, & x \neq 0 \\ k , & x = 0\end{cases}$  is continuous at x = 0, then write the value of k.

Q 5 | Page 41

If the function   $f\left( x \right) = \frac{\sin 10x}{x}, x \neq 0$ is continuous at x = 0, find f (0).

Q 6 | Page 42

If $f\left( x \right) = \begin{cases}\frac{x^2 - 16}{x - 4}, & if x \neq 4 \\ k , & if x = 4\end{cases}$  is continuous at x = 4, find k.

Q 7 | Page 42

Determine whether $f\left( x \right) = \binom{\frac{\sin x^2}{x}, x \neq 0}{0, x = 0}$  is continuous at x = 0 or not.

Q 8 | Page 42

If  $f\left( x \right) = \binom{\frac{1 - \cos x}{x^2}, x \neq 0}{k, x = 0}$  is continuous at x = 0, find k

Q 9 | Page 42

If $f\left( x \right) = \begin{cases}\frac{\sin^{- 1} x}{x}, & x \neq 0 \\ k , & x = 0\end{cases}$is continuous at x = 0, write the value of k.

Q 10 | Page 42

Write the value of b for which $f\left( x \right) = \begin{cases}5x - 4 & 0 < x \leq 1 \\ 4 x^2 + 3bx & 1 < x < 2\end{cases}$  is continuous at x = 1.

Q 11 | Page 42

Determine the value of the constant 'k' so that function

$\left( x \right) = \begin{cases}\frac{kx}{\left| x \right|}, & if x < 0 \\ 3 , & if x \geq 0\end{cases}$  is continuous at

Pages 42 - 47

Q 1 | Page 42

The function

$f\left( x \right) = \frac{4 - x^2}{4x - x^3}$

(a) discontinuous at only one point
(b) discontinuous exactly at two points
(c) discontinuous exactly at three points
(d) none of these

Q 2 | Page 42

If f (x) = | x − a | ϕ (x), where ϕ (x) is continuous function, then
(a) f' (a+) = ϕ (a) (b) f' (a) = −ϕ (a)  (c) f' (a+) = f' (a)
(d) none of these

Q 3 | Page 42

If $f\left( x \right) = \left| \log_{10} x \right|$ then at x = 1
(a) f (x) is continuous and f' (1+) = log10 e
(b) f (x) is continuous and f' (1+) = log10 e
(c) f (x) is continuous and f' (1) = log10 e
(d) f (x) is continuous and f' (1) = −log10 e

Q 4 | Page 42

If  $f\left( x \right) = \begin{cases}\frac{{36}^x - 9^x - 4^x + 1}{\sqrt{2} - \sqrt{1 + \cos x}}, & x \neq 0 \\ k , & x = 0\end{cases}$is continuous at x = 0, then k equals

(a) $16\sqrt{2}$ log 2 log 3

(b)$16\sqrt{2}$

(c)$16\sqrt{2}$  ln 2 ln 3

(d) none of these

Q 5 | Page 43
then f (x) is continuous for all
$f\left( x \right) = \begin{cases}\frac{\left| x^2 - x \right|}{x^2 - x}, & x \neq 0, 1 \\ 1 , & x = 0 \\ - 1 , & x = 1\end{cases}$ (a) (b) x except at x = 0 (c) x except at x = 1 (d) x except at x = 0 and x = 1.
Q 6 | Page 43

If $f\left( x \right) = \begin{cases}\frac{1 - \sin x}{\left( \pi - 2x \right)^2} . \frac{\log \sin x}{\log\left( 1 + \pi^2 - 4\ pix + 4 x^2 \right)}, & x \neq \frac{\pi}{2} \\ k , & x = \frac{\pi}{2}\end{cases}$is continuous at x = π/2, then k =

(a)  $- \frac{1}{16}$  (b) $- \frac{1}{32}$  (c)  $- \frac{1}{64}$ (d) $- \frac{1}{28}$

Q 7 | Page 43

If f (x) = (x + 1)cot x be continuous at x = 0, then f (0) is equal to (a) 0 (b) 1/(c) e (d) none of these

Q 8 | Page 43

If  $f\left( x \right) = \begin{cases}\frac{\log\left( 1 + ax \right) - \log\left( 1 - bx \right)}{x}, & x \neq 0 \\ k , & x = 0\end{cases}$ and f (x) is continuous at x = 0, then the value of k is (a) a − (b) a + (c) log a + log (d) none of these

Q 9 | Page 43

The function  $f\left( x \right) = \begin{cases}\frac{e^{1/x} - 1}{e^{1/x} + 1}, & x \neq 0 \\ 0 , & x = 0\end{cases}$

(a) is continuous at x = 0 (b) is not continuous at x = 0
(c) is not continuous at x = 0, but can be made continuous at x = 0 (d) none of these

Q 10 | Page 43

Let  $f\left( x \right) = \left\{ \begin{array}\frac{x - 4}{\left| x - 4 \right|} + a, & x < 4 \\ a + b , & x = 4 \\ \frac{x - 4}{\left| x - 4 \right|} + b, & x > 4\end{array} . \right.$Then, f (x) is continuous at x = 4 when
(a) a = 0, b = 0  (b) a = 1, b = 1 (c) a = −1, b = 1
(d) a = 1, b = −1.

Q 11 | Page 43

If the function  $\Rightarrow b = - 1 and a = 1$

is continuous at x = 0, then the value of k is
(a) 0 (b) 1 (c) −1 (d) e.

Q 12 | Page 43

Let f (x) = | x | + | x − 1|, then
(a) f (x) is continuous at x = 0, as well as at x = 1
(b) f (x) is continuous at x = 0, but not at x = 1
(c) (x) is continuous at x = 1, but not at x = 0
(d) none of these

Q 13 | Page 44

Let  $f\left( x \right) = \begin{cases}\frac{x^4 - 5 x^2 + 4}{\left| \left( x - 1 \right) \left( x - 2 \right) \right|}, & x \neq 1, 2 \\ 6 , & x = 1 \\ 12 , & x = 2\end{cases}$ (a) R
(b) R − {1} (c) R − {2} (d) − {1, 2}

Q 14 | Page 44

If  $f\left( x \right) = \begin{cases}\frac{\sin (a + 1) x + \sin x}{x} , & x < 0 \\ c , & x = 0 \\ \frac{\sqrt{x + b x^2} - \sqrt{x}}{bx\sqrt{x}} , & x > 0\end{cases}$is continuous at x = 0, then (a) a =  $- \frac{3}{2}$ , b = 0, c = $\frac{1}{2}$ (b) a = $- \frac{3}{2}$ , b = 1, c =

$- \frac{1}{2}$ (c) a =$- \frac{3}{2}$, b ∈ R − {0}, c =
$\frac{1}{2}$  (d) none of these
Q 15 | Page 44

If $f\left( x \right) = \begin{cases}mx + 1 , & x \leq \frac{\pi}{2} \\ \sin x + n, & x > \frac{\pi}{2}\end{cases}$ is continuous at $x = \frac{\pi}{2}$ (a) m = 1, n = 0 (b)  $m = \frac{n\pi}{2} + 1$ (c)  $n = \frac{m\pi}{2}$ (d) $m = n = \frac{\pi}{2}$

Q 16 | Page 44

The value of f (0), so that the function

$f\left( x \right) = \frac{\sqrt{a^2 - ax + x^2} - \sqrt{a^2 + ax + x^2}}{\sqrt{a + x} - \sqrt{a - x}}$ (a) a3/2  (b) a1/2  (c) −a1/2 (d) −a3/2
Q 17 | Page 44

The function  $f\left( x \right) = \begin{cases}1 , & \left| x \right| \geq 1 & \\ \frac{1}{n^2} , & \frac{1}{n} < \left| x \right| & < \frac{1}{n - 1}, n = 2, 3, . . . \\ 0 , & x = 0 &\end{cases}$ (a) is discontinuous at finitely many points (b) is continuous everywhere (c) is discontinuous only at  $x = \pm \frac{1}{n}$n ∈ Z − {0} and x = 0 (d) none of these

Q 18 | Page 44
The value of f (0), so that the function

$f\left( x \right) = \frac{\left( 27 - 2x \right)^{1/3} - 3}{9 - 3 \left( 243 + 5x \right)^{1/5}}\left( x \neq 0 \right)$ is continuous, is given by  (a) $\frac{2}{3}$

(b) 6 (c) 2  (d) 4

Q 19 | Page 44

The value of f (0) so that the function

$f\left( x \right) = \frac{2 - \left( 256 - 7x \right)^{1/8}}{\left( 5x + 32 \right)^{1/5} - 2},$  0 is continuous everywhere, is given by
(a) −1 (b) 1 (c) 26 (d) none of these
Q 20 | Page 45
$f\left( x \right) = \begin{cases}\frac{\sqrt{1 + px} - \sqrt{1 - px}}{x}, & - 1 \leq x < 0 \\ \frac{2x + 1}{x - 2} , & 0 \leq x \leq 1\end{cases}$is continuous in the interval [−1, 1], then p is equal to
(a) −1 (b) −1/2 (c) 1/2 (d) 1

Q 21 | Page 45

The function

$f\left( x \right) = \begin{cases}x^2 /a , & 0 \leq x < 1 \\ a , & 1 \leq x < \sqrt{2} \\ \frac{2 b^2 - 4b}{x^2}, & \sqrt{2} \leq x < \infty\end{cases}$is continuous for 0 ≤ x < ∞, then the most suitable values of a and b are
(a) a = 1, b = −1 (b) a = −1, b = 1 +
$\sqrt{2}$ (c) a = −1, b = 1 (d) none of these

Q 22 | Page 45

If  $f\left( x \right) = \frac{1 - \sin x}{\left( \pi - 2x \right)^2},$ when x ≠ π/2 and f (π/2) = λ, then f (x) will be continuous function at x= π/2, where λ =
(a) 1/8 (b) 1/4 (c) 1/2 (d) none of thes

Q 23 | Page 45

The value of a for which the functionv

$f\left( x \right) = \begin{cases}\frac{\left( 4^x - 1 \right)^3}{\sin\left( x/a \right) \log \left\{ \left( 1 + x^2 /3 \right) \right\}}, & x \neq 0 \\ 12 \left( \log 4 \right)^3 , & x = 0\end{cases}$may be continuous at x = 0 is (a) 1 (b) 2 (c) 3 (d) none of these

Q 24 | Page 45

The function f (x) = tan x is discontinuous on the set
(a) {n π : n ∈ Z}
(b) {2n π : n ∈ Z}
(c) $\left\{ \left( 2n + 1 \right)\frac{\pi}{2}: n \in Z \right\}$

(d) $\left\{ \frac{n\pi}{2}: n \in Z \right\}$

Q 25 | Page 45

The function

$f\left( x \right) = \begin{cases}\frac{\sin 3x}{x}, & x \neq 0 \\ \frac{k}{2} , & x = 0\end{cases}$  is continuous at x = 0, then k = (a) 3
(b) 6 (c) 9 (d) 12
Q 26 | Page 45

If the function

$f\left( x \right) = \frac{2x - \sin^{- 1} x}{2x + \tan^{- 1} x}$ is continuous at each point of its domain, then the value of f (0) is (a) 2 (b)  $\frac{1}{3}$ (c)  $- \frac{1}{3}$  (d) $\frac{2}{3}$
Q 27 | Page 45

The value of b for which the function

$f\left( x \right) = \begin{cases}5x - 4 , & 0 < x \leq 1 \\ 4 x^2 + 3bx , & 1 < x < 2\end{cases}$ is continuous at every point of its domain, is (a) −1  (b) 0  (c) $\frac{13}{3}$  (d) 1
Q 28 | Page 45

If  $f\left( x \right) = \frac{1}{1 - x}$ , then the set of points discontinuity of the function f (f(f(x))) is(a) {1}
(b) {0, 1}  (c) {−1, 1} (d) none of these

Q 29 | Page 46

Let  $f\left( x \right) = \frac{\tan\left( \frac{\pi}{4} - x \right)}{\cot 2x}, x \neq \frac{\pi}{4} .$  The value which should be assigned to f (x) at  $x = \frac{\pi}{4},$so that it is continuous everywhere is (a) 1 (b) 1/2 (c) 2
(d) none of these

Q 30 | Page 46

The function  $f\left( x \right) = \frac{x^3 + x^2 - 16x + 20}{x - 2}$ is not defined for x = 2. In order to make f (x) continuous at x = 2, Here f (2) should be defined as

(a) 0 (b) 1 (c) 2 (d) 3

Q 31 | Page 46

If  $f\left( x \right) = \begin{cases}a \sin\frac{\pi}{2}\left( x + 1 \right), & x \leq 0 \\ \frac{\tan x - \sin x}{x^3}, & x > 0\end{cases}$ (a)  $\frac{1}{2}$  (b) $\frac{1}{3}$  (c) $\frac{1}{4}$ (d)  $\frac{1}{6}$

Q 32 | Page 46

If  $f\left( x \right) = \left\{ \begin{array}a x^2 + b , & 0 \leq x < 1 \\ 4 , & x = 1 \\ x + 3 , & 1 < x \leq 2\end{array}, \right.$ then the value of (ab) for which f (x) cannot be continuous at x = 1, is
(a) (2, 2) (b) (3, 1) (c) (4, 0) (d) (5, 2)

Q 33 | Page 46

If the function f (x) defined by

$f\left( x \right) = \begin{cases}\frac{\log \left( 1 + 3x \right) - \log \left( 1 - 2x \right)}{x}, & x \neq 0 \\ k , & x = 0\end{cases}$ is continuous at x = 0, then k =

(a) 1 (b) 5 (c) −1 (d) none of these

Q 34 | Page 46

If $f\left( x \right) = \begin{cases}\frac{1 - \cos 10x}{x^2} , & x < 0 \\ a , & x = 0 \\ \frac{\sqrt{x}}{\sqrt{625 + \sqrt{x}} - 25}, & x > 0\end{cases}$ then the value of a so that f (x) may be continuous at x = 0, is (a) 25
(b) 50 (c) −25 (d) none of these

Q 35 | Page 46

If  $f\left( x \right) = x \sin\frac{1}{x}, x \neq 0,$then the value of the function at = 0, so that the function is continuous at x = 0, is(a) 0 (b) −1 (c) 1 (d) indeterminate

Q 36 | Page 46

The value of k which makes  $f\left( x \right) = \begin{cases}\sin\frac{1}{x}, & x \neq 0 \\ k , & x = 0\end{cases}$    continuous at x = 0, is

(a) 8 (b) 1 (c) −1 (d) none of these

Q 37 | Page 46

The values of the constants ab and c for which the function  $f\left( x \right) = \begin{cases}\left( 1 + ax \right)^{1/x} , & x < 0 \\ b , & x = 0 \\ \frac{\left( x + c \right)^{1/3} - 1}{\left( x + 1 \right)^{1/2} - 1}, & x > 0\end{cases}$ may be continuous at x = 0, are

(a) $a = \log_e \left( \frac{2}{3} \right), b = - \frac{2}{3}, c = 1$

(b) $a = \log_e \left( \frac{2}{3} \right), b = \frac{2}{3}, c = - 1$

(c)  $a = \log_e \left( \frac{2}{3} \right), b = \frac{2}{3}, c = 1$

(d) none of these

Q 38 | Page 47

The points of discontinuity of the function

$f\left( x \right) = \begin{cases}2\sqrt{x} , & 0 \leq x \leq 1 \\ 4 - 2x , & 1 < x < \frac{5}{2} \\ 2x - 7 , & \frac{5}{2} \leq x \leq 4\end{cases}is \left( are \right)$ (a) x = 1, $x = \frac{5}{2}$ (b) $x = \frac{5}{2}$ (c)  $x = 1, \frac{5}{2}, 4$  (d) x = 0, 4

Q 39 | Page 47

If  $f\left( x \right) = \begin{cases}\frac{1 - \sin^2 x}{3 \cos^2 x} , & x < \frac{\pi}{2} \\ a , & x = \frac{\pi}{2} \\ \frac{b\left( 1 - \sin x \right)}{\left( \pi - 2x \right)^2}, & x > \frac{\pi}{2}\end{cases}$. Then, f (x) is continuous at  $x = \frac{\pi}{2}$

(a)  $a = \frac{1}{3},$ (b) $a = \frac{1}{3}, b = \frac{8}{3}$ (c)  $a = \frac{2}{3}, b = \frac{8}{3}$(d) none of these

Q 40 | Page 47

The points of discontinuity of the function$f\left( x \right) = \begin{cases}\frac{1}{5}\left( 2 x^2 + 3 \right) , & x \leq 1 \\ 6 - 5x , & 1 < x < 3 \\ x - 3 , & x \geq 3\end{cases}is \left( are \right)$  (a) x = 1
(b) x = 3  (c) x = 1, 3 (d) none of these

Q 41 | Page 47

The value of a for which the function $f\left( x \right) = \begin{cases}5x - 4 , & if 0 < x \leq 1 \\ 4 x^2 + 3ax, & if 1 < x < 2\end{cases}$ is continuous at every point of its domain, is (a)  $\frac{13}{3}$ (b) 1 (c) 0
(d) −1

Q 42 | Page 47

If  $f\left( x \right) = \begin{cases}\frac{\sin \left( \cos x \right) - \cos x}{\left( \pi - 2x \right)^2}, & x \neq \frac{\pi}{2} \\ k , & x = \frac{\pi}{2}\end{cases}$is continuous at x = π/2, then k is equal to (a) 0
(b)  $\frac{1}{2}$ (c) 1 (d) −1

RD Sharma solutions for Class 12 Mathematics chapter 9 - Continuity

RD Sharma solutions for Class 12 Maths chapter 9 (Continuity) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 9 Continuity are Derivative - Exponential and Log, Concept of Differentiability, Proof Derivative X^n Sin Cos Tan, Infinite Series, Higher Order Derivative, Algebra of Continuous Functions, Continuous Function of Point, Mean Value Theorem, Second Order Derivative, Derivatives of Functions in Parametric Forms, Logarithmic Differentiation, Exponential and Logarithmic Functions, Derivatives of Implicit Functions, Derivatives of Inverse Trigonometric Functions, Derivatives of Composite Functions - Chain Rule, Concept of Continuity.

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