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CBSE Mathematics for Class XII

Dinesh Khattar; Anita Khattar

Chapter 5

Continuity and Differentiability - all with Video Answers

Educators

Section 1

Continuity

02:29

Problem 1

Which of the following functions are continuous at the indicated points:
(i) $f(x)=\sin ^{2} x+x^{2}-2 x$ at $x=0$
(ii) $f(x)=\left\{\begin{array}{ll}x, & \text { if } x<0 \\ x^{2}, & \text { if } x \geq 0\end{array}\right.$ at $x=1$
(iii) $f(x)=\left\{\begin{array}{ll}5 x-4, & \text { if } x \leq 1 \\ 4 x^{2}-3 x, & \text { if } x>0\end{array}\right.$ at $x=1$
(iv) $f(x)=\left\{\begin{array}{ll}2 x-1, & \text { if } x<0 \\ 2 x+1, & \text { if } x \geq 0\end{array}\right.$ at $x=0$
(v) $f(x)=\left\{\begin{array}{ll}\frac{\sin x}{x}, & \text { if } x<0 \\ x+1, & \text { if } x \geq 0\end{array}\right.$ at $x=0$
(vi) $f(x)=\left\{\begin{array}{ll}\frac{x}{|x|}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right.$ at $x=0$
(vii) $f(x)=\left\{\begin{array}{ll}2 x-1, & \text { if } x<0 \\ 2 x+6, \text { if } x \geq 0\end{array}\right.$ at $x=0$
(viii) $f(x)=\left\{\begin{array}{ll}x^{2} \sin \frac{1}{x}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right.$ at $x=0$
(ix) $f(x)=\left\{\begin{array}{ll}\frac{\sin x}{x}+\cos x, & \text { if } x \neq 0 \\ 2, & \text { if } x=0\end{array}\right.$ at $x=0$
(x) $f(x)=\left\{\begin{array}{ll}\frac{x^{2}-1}{x+1}, & \text { if } x \neq-1 \\ -2, & \text { if } x=-1\end{array}\right.$ at $x=-1$
(xi) $f(x)=\left\{\begin{array}{ll}\frac{x^{2}-x-6}{x^{2}-2 x-3}, & \text { if } x \neq 3 \\ \frac{5}{3}, & \text { if } x=3\end{array}\right.$ at $x=3$
(xii) $f(x)=\left\{\begin{array}{ll}2 x-1, & \text { if } x<2 \\ \frac{3 x}{2}, & \text { if } x \geq 2\end{array}\right.$ at $x=2$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:37

Problem 2

Examine the continuity of the following functions at indicated points:
(i) $f(x)=\left\{\begin{array}{ll}\frac{x}{\sin 3 x}, & x \neq 0 \\ 1, & x=0\end{array}\right.$ at $x=0$
(ii) $f(x)=\left\{\begin{array}{ll}\frac{\sin 3 x}{x}, & x \neq 0 \\ 1, & x=0\end{array}\right.$ at $x=0 .$
(iii) $f(x)=\left\{\begin{array}{ll}\frac{\sin 2 x}{\sin 3 x}, & x \neq 0 \\ 2, & x=0\end{array}\right.$ at $x=0$.
(iv) $f(x)=\left\{\begin{array}{ll}\frac{1-\cos 4 x}{x^{2}}, & x \neq 0 \\ 8, & x=0\end{array}\right.$ at $x=0$
(v) $f(x)=\left\{\begin{array}{ll}\frac{\cos t}{\frac{\pi}{2}-t}, & t \neq \frac{\pi}{2} \\ 1, & t=\frac{\pi}{2}\end{array}\right.$ at $t=\frac{\pi}{2} .$
(vi) $f(x)=\left\{\begin{array}{ll}\frac{|x-a|}{x-a}, & x \neq 0 \\ 1, & x=0\end{array}\right.$ at $x=a$
(vii) $f(x)=\left\{\begin{array}{ll}|x|, & x \leq 2 \\ {[x],} & x>2\end{array}\right.$ at $x=2$
(viii) $f(x)= \begin{cases}\frac{[x]-1}{x-1}, & x \neq 1 \text { at } x=1 \\ -1, & x=1\end{cases}$

Tanishq Gupta
Tanishq Gupta
Numerade Educator
01:32

Problem 3

In each of the following. find the value of the constant $k$ so that the given function is continuous at the indicated point:
(i) $f(x)=\left\{\begin{array}{ll}\frac{k \cos x}{\pi-2 x}, & \text { if } x \neq \frac{\pi}{2} \\ 3, & \text { if } x=\frac{\pi}{2}\end{array}\right.$ at $x=\frac{\pi}{2}$
(ii) $f(x)=\left\{\begin{array}{ll}k x^{2}, & \text { if } x \leq 2 \\ 3, & \text { if } x>2\end{array}\right.$ at $x=2$
(iii) $f(x)=\left\{\begin{array}{ll}k\left(x^{2}-2 x\right), & \text { if } x<0 \\ \cos x, & \text { if } x \geq 0\end{array}\right.$ at $x=0$
(iv) $f(x)=\left\{\begin{array}{ll}\frac{1-\cos 2 x}{2 x^{2}}, & x \neq 0 \\ k, & x=0\end{array}\right.$ continuous at $x=0$
(v) $f(x)=\left\{\begin{array}{cl}\frac{\sin x+x \cos x}{x}, & x \neq 0 \\ k, & x=0\end{array}\right.$ continuous at $x=0$
(vi) $f(x)=\left\{\begin{array}{ll}\frac{\sin 5 x}{3 x}, & x \neq 0 \\ k, & x=0\end{array}\right.$ continuous at $x=0$
(vii) $f(x)=\left\{\begin{array}{ll}\frac{x^{2}-4}{x-2}, & x \neq 2 \\ k, & x=2\end{array}\right.$ continuous at $x=2$
(viii) $f(x)= \begin{cases}k\left(x^{2}-2 x\right), & x \leq 0 \\ 4 x+1, & x>0\end{cases}$
(ix) $f(x)=\left\{\begin{array}{ll}k x+1, & x \leq \pi \\ \cos x, & x>\pi\end{array}\right.$ at $x=\pi$
(x) $f(x)=\left\{\begin{array}{ll}k x+1, & x \leq 5 \\ 3 x-5, & x>5\end{array}\right.$ at $x=5$

Manisha Sarker
Manisha Sarker
Numerade Educator
02:49

Problem 4

Show that the function $f$ given by $f(x)=|x|+|x-1|, x \in R$ is continuous both at $x=0$ and $x=1$

Eleni Katirtzoglou
Eleni Katirtzoglou
Numerade Educator
03:16

Problem 5

(i) Discuss the continuity of the function $f$ given by $f(x)=|x-1|+|x-2|$ at $x=1$ and $x=2 .$
(ii) Discuss the continuity of the function $f(x)=|x+1|+|x+1|$ at $x=-1$ and $x=1$.

Gaurav Kumar
Gaurav Kumar
Numerade Educator
02:12

Problem 6

Discuss continuity of the function $f$ given by $f(x)=\left\{\begin{array}{ll}1, & x \neq 0 \\ 2, & x=0\end{array}\right.$ at $x=0$.

Tanishq Gupta
Tanishq Gupta
Numerade Educator
03:16

Problem 7

Discuss the continuity at $x=0$ for the function $f(x)=2 x-|x|$

Gaurav Kumar
Gaurav Kumar
Numerade Educator
02:30

Problem 8

Show that the function $\frac{\sin x}{|x|}$ is discontinuous at $x=0$.

Gaurav Kumar
Gaurav Kumar
Numerade Educator
02:21

Problem 9

Prove that the function $f(x)=\left\{\begin{array}{ll}\frac{x}{|x|+2 x^{2}}, & x \neq 0 \\ k, & x=0\end{array}\right.$ remains discontinuous at $x=0$, regardless the choice of $k$.

Dwijendra Rao
Dwijendra Rao
Numerade Educator
03:06

Problem 10

Is the function defined by $x^{2}-\sin x+5$ continuous at $x=\pi$ ?

Manisha Sarker
Manisha Sarker
Numerade Educator
01:31

Problem 11

Determine the value of constant $k$ so that the function $f(x)= \begin{cases}\frac{1-\cos k x}{x \sin x}, & x \neq 0 \\ 2, & x=0\end{cases}$
is continuous at $x=0$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:29

Problem 12

(i) Find $f(0)$ if the function $f(x)=\frac{x+\sin x}{3 x+2 \tan x}$, for $x \neq 0$ is continuous at $x=0$.
(ii) Find $f(0)$ if the function $f(x)=\frac{\sin x}{x}$ is continuous at $x=0$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:35

Problem 13

Discuss the continuity of the following functions at the indicated points:
(i) $f(x)=\left\{\begin{array}{cc}\frac{x-|x|}{x}, & x \neq 0 \\ 2, & x=0\end{array}\right.$ at $x=0$
(ii) $f(x)=\left\{\begin{array}{cl}\frac{e^{x}-1}{\log (1+3 x)}, & x \neq 0 \\ 5, & x=0\end{array}\right.$ at $x=0$
(iii) $f(x)=\left\{\begin{array}{cc}x \sin \frac{1}{x}, & x \neq 0 \\ 2, & x=0\end{array}\right.$ at $x=0$

Gaurav Kumar
Gaurav Kumar
Numerade Educator
01:31

Problem 14

For what value of $k, f(x)=\left\{\begin{array}{cl}{[\cos x],} & x \neq 0 \\ k, & x=0\end{array}\right.$ is continuous at $x=0$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:30

Problem 15

If $f(x)=\left\{\begin{array}{cl}3-2 x, & 0<x<2 \\ 4 x^{2}-3 a x, & 2 \leq x<5\end{array}\right.$ is continuous in its domain, find the value of $a$.

Amy Jiang
Amy Jiang
Numerade Educator
04:23

Problem 16

For what value(s) of $a$ and $b$
$$
f(x)=\left\{\begin{aligned}
\frac{x+2}{|x+2|}+a, & x<-2 \\
a+b, & x=-2 \\
\frac{x+2}{|x+2|}+b, & x>-2
\end{aligned} \text { is continuous at } x=2 .\right.
$$

John Irizar
John Irizar
Numerade Educator
01:45

Problem 17

If $f(x)=\frac{1}{1-x}$, find the points of discontinuity of $f[f\{f(x)\}]$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:03

Problem 18

For what value of $k, f(x)=\left\{\begin{array}{cl}\frac{3 x-\tan x}{5 x-\sin x}, & x>2 \\ k, & x=2 \text { is continuous at } x=0 \text {. } \\ 3 x^{2}-4 x+\frac{1}{2}, & x<2\end{array}\right.$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:30

Problem 19

Show that $f(x)=x-[x]$ is discontinuous at $x=2$. Also, discuss the continuity at $x=2.5$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator