KC Sinha Mathematics Solution Class 11 Chapter 28 Limit at Infinity and Infinite Limits and Existence of Limits Exercise 28.2

 Exercise 28.2

Page no 28.20

Question 1

If $f(x)=x^{2}+x+2$, when $x<1$

$=x^{4}+3, \quad \text { when } x>1$

then does $\lim _{x \rightarrow 1} f(x)$ exist ?

Question 2

If $f(x)=x^{3}$, when $x<-1$ $=x^{5}$, when $x>-1$ $=-1$, when $x=-1$ then find $\lim _{x \rightarrow-1} f(x)$.

Question 3

Find $\lim _{x \rightarrow 2} \phi(x)$, where

$\begin{aligned}\phi(x) &=5 ; \text { when } x \neq 2 \\&=3, \text { when } x=2\end{aligned}$

Question 4

(i) Evaluate $\lim _{x \rightarrow 0} f(x)$, where

$f(x)=\left\{\begin{array}{cc}\frac{x}{|x|}, & x \neq 0 \\ 0, & x=0\end{array}\right.$

(ii) Evaluate $\lim _{x \rightarrow 0} f(x)$, where

$f(x)=\left\{\begin{array}{cc}\frac{|x|}{x}, & x \neq 0 \\ 0, & x=0\end{array}\right.$

Question 5

Prove that $\lim _{x \rightarrow 1} \frac{x^{2}-1}{|x-1|}$ does not exist.

Question 6

If $f(x)=\left\{\begin{array}{cc}4, & x \geq 3 \\ x+1, & x<3\end{array}\right.$

Find $\lim _{x \rightarrow 3} f(x)$

Question 7

(i) Let $f(x)=\left\{\begin{array}{ll}x+1, & x \geq 0 \\ x-1, & x<0\end{array}\right.$, prove that $\lim _{x \rightarrow 0} f(x)$ does not exist.

(ii) Let $f(x)=\left\{\begin{array}{cc}x^{2}-1, & x \leq 1 \\ -x^{2}-1, & x>1\end{array}\right.$ prove that $\lim _{x \rightarrow 1} f(x)$ does not exist.

Question 8

Evaluate the following one sided limits :

(i) $\lim _{x \rightarrow 2-0}[x]$

(ii) $\lim _{x \rightarrow-5^{+}}[-x]$

(iii) $\lim _{x \rightarrow 0^{-}} \frac{\left|x^{3}\right|}{|x|}$

(iv) $\lim _{x \rightarrow 3^{-}} x[x]$

Page no 28.21

Question 9

Evaluate the following one sided limits :

(i) $\lim _{x \rightarrow 0^{-}}(2-\cot x)$

(ii) $\lim _{x \rightarrow-\frac{\pi}{2}+}(\sec x)$

(iii) $\lim _{x \rightarrow 0^{-}}(1+\operatorname{cosec} x)$

Question 10

Evaluate the following one sided limits :

(i) $\lim _{x \rightarrow 2^{+}} \frac{x-3}{x^{2}-4}$

(ii) $\lim _{x \rightarrow 0^{+}} \frac{1}{3 x}$

(iii) $\lim _{x \rightarrow 2^{-}} \frac{x-3}{x^{2}-4}$

(iv) $\lim _{x \rightarrow-8^{+}} \frac{2 x}{x+8}$

(v) $\lim _{x \rightarrow 0^{+}} \frac{2}{x^{1 / 5}}$

Question 11

Find $\lim _{x \rightarrow 5^{+}} f(x)$ and $\lim _{x \rightarrow 5^{-}} f(x)$, where $f(x)=|x|-5$

Question 12

Find $\lim _{x \rightarrow 0} f(x)$ and $\lim _{x \rightarrow 1} f(x)$, where $f(x)=\left\{\begin{array}{cc}2 x+3, & x \leq 0 \\ 3(x+1), & x>0\end{array}\right.$

Question 13

Find the following limits if they exist :

(i) $\lim _{x \rightarrow 1}[x]$

(ii) $\lim _{x \rightarrow \frac{5}{2}}[x]$

(iii) $\lim _{x \rightarrow \frac{7}{3}}[-x]$

Question 14

 (i) Find $\lim _{x \rightarrow-1-0} \frac{\sqrt{x^{2}-5 x-6}}{x+3}$

(ii) Find $\lim _{x \rightarrow 2^{-}} \frac{x^{2}-4 x+4}{x-2}$

(iii) Find $\lim _{x \rightarrow-3^{+}} \frac{[x-7]}{[x+4]}$

Question 15

If 

$\begin{aligned} f(x) &=\frac{\sin 3 x}{x}, \text { when } x<0 \\ &=\frac{\tan b x}{x}, \text { when } x>0 \text { and } \lim _{x \rightarrow 0} f(x) \text { exists, find the value of } b \text {. } \end{aligned}$

Question 16

Let $f(x)=\left\{\begin{array}{cc}x+2, & x \leq-1 \\ c x^{2}, & x>-1\end{array}\right.$; Find $c$ if $\lim _{x \rightarrow-1} f(x)$ exists.