KC Sinha Mathematics Solution Class 11 Chapter 27 Limits of a Function Exercise 27.1

 Exercise 27.1

Page no 27.21

Question 1

Evaluate the following limits :

(i) $\lim _{x \rightarrow 1}(x-2)$

(ii) $\lim _{x \rightarrow 2}(x+2)$

(iii) $\lim _{x \rightarrow 0}(9)$

(iv) $\lim _{x \rightarrow 3} \frac{x^{2}-9}{x+2}$

(v) $\lim _{x \rightarrow 2}(3-x)$

(vi) $\lim _{x \rightarrow 2} x(x-1)$

(vii) $\lim _{x \rightarrow-1} \frac{x^{3}-3 x+1}{x-1}$

(viii) $\lim _{x \rightarrow 2} \frac{x^{2}+4}{x+2}$

Page no 27.22

(ix) $\lim _{x \rightarrow 1} \frac{x^{2}+5}{x-2}$

(x) $\lim _{x \rightarrow 0} \frac{a x+b}{c x+d}, d \neq 0$

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

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

(xiii) $\lim _{r \rightarrow 1} \pi r^{2}$

(xiv) $\lim _{x \rightarrow 3} x(x+3)$

(xv) $\lim _{x \rightarrow 3} x+3$

(xvi) $\lim _{x \rightarrow 1}\left(x^{3}-x^{2}+1\right)$

(xvii) $\lim _{x \rightarrow \pi}\left(x-\frac{22}{7}\right)$

(xviii) $\lim _{x \rightarrow-1}\left(1+x+x^{2}+\ldots+x^{10}\right)$

(xix) $\lim _{x \rightarrow 0} \frac{a x+b}{c x+1}$

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

(xxi) $\lim _{x \rightarrow 1}\left(\frac{x^{2}+1}{x+100}\right)$

(xxii) $\lim _{x \rightarrow 1} \frac{a x^{2}+b x+c}{c x^{2}+b x+a}, a+b+c \neq 0$

(xxiii) $\lim _{x \rightarrow-1} \frac{x^{10}+x^{5}+1}{x-1}$

Question 2

Evaluate $\lim _{x \rightarrow 1} \frac{x^{3}-1}{x-1}$

Question 3

Find $\lim _{x \rightarrow a} \frac{x^{4}-a^{4}}{x-a}$

Question 4

Find $\lim _{x \rightarrow 2} \frac{x^{4}-16}{x-2}$

Question 5

Find $\lim _{t \rightarrow-3} \frac{x^{3}+27}{x^{5}+243}$

Question 6

Find (i) $\lim _{t \rightarrow 1} \frac{\sqrt{t}-1}{\sqrt[3]{t}-1}$

Question 7

Find $\lim _{x \rightarrow a} \frac{x-a}{x^{3 / 2}-a^{3 / 2}}$

Question 8

Find $\lim _{x \rightarrow 1} \frac{x-1}{x^{1 / 4}-1}$

Question 9

Find $\lim _{x \rightarrow 2} \frac{\sqrt[3]{x-1}-1}{x-2}$

Question 10

Find $\lim _{x \rightarrow 0} \frac{(1+x)^{n}-1}{x}$

Question 11

Find $\lim _{x \rightarrow 1} \frac{x^{5}-1}{x-1}$

Question 12

Find $\lim _{x \rightarrow 0} \frac{(1+x)^{1 / n}-1}{x}$

Question 13

Find

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

Question 14

Find $\lim _{x \rightarrow 1}\left(\frac{1-x^{-1 / 3}}{1-x^{-2 / 3}}\right)$

Question 15

Find $\lim _{x \rightarrow 0} \frac{(1+x)^{5}-1}{x}$

Question 16

Find $\lim _{x \rightarrow a} \frac{(x+2)^{5 / 2}-(a+2)^{5 / 2}}{x-a}$

Question 17

Find the positive integer $n$ so that $\lim _{x \rightarrow 3} \frac{x^{n}-3^{n}}{x-3}=108$

Question 18

Find all possible values of $n \in N$ if $\lim _{x \rightarrow 2} \frac{x^{n}-2^{n}}{x-2}=80$

Question 19

Find all possible values of $a$, if $\lim _{x \rightarrow a} \frac{x^{3}-a^{3}}{x-a}=\lim _{x \rightarrow 1} \frac{x^{4}-1}{x-1}$

Page no 27.23

Question 20

$\operatorname{Find}_{x \rightarrow 0} \frac{\sqrt{1+4 x}-\sqrt{1-3 x}}{x}$

Question 21

Find $\lim _{x \rightarrow 0} \frac{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}{x}$

Question 22

Find $\lim _{x \rightarrow 1} \frac{\sqrt{2-x}-1}{1-x}$

Question 23

Find $\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1}{x}$

Question 24

Find $\lim _{x \rightarrow 0} \frac{2-\sqrt{4-x^{2}}}{x^{2}}$

Question 25

Find $\lim _{x \rightarrow 0} \frac{a-\sqrt{a^{2}-x^{2}}}{x}$

Question 26

Find $\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-\sqrt{1-x}}{x}$

Question 27

Find $\lim _{h \rightarrow 0} \frac{\sqrt{x+h}-\sqrt{x}}{h}, x \neq 0$

Question 28

Find $\lim _{x \rightarrow 0} \frac{\sqrt{a+x}-\sqrt{a}}{x \sqrt{a(a+x)}}$

Question 29

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

Question 30

Find $\lim _{x \rightarrow 2} \frac{x^{2}-4}{\sqrt{x+2}-\sqrt{3 x-2}}$

Question 31

Find $\lim _{x \rightarrow 1} \frac{x-1}{\sqrt{x^{2}-1}+\sqrt{x-1}}$

Question 32

Find $\lim _{x \rightarrow 0} \frac{\sqrt{1+x^{2}}-\sqrt{1+x}}{\sqrt{x^{2}+1}+\sqrt{1+x}}$

Question 33

Find $\lim _{x \rightarrow 0} \frac{\sqrt{1+2 x^{2}}-\sqrt{1+x}}{\sqrt{4+4 x^{2}}-\sqrt{4-3 x}}$

Question 34

Find $\lim _{x \rightarrow 1} \frac{\sqrt{x^{2}+8}-\sqrt{10-x^{2}}}{\sqrt{x^{2}+3}-\sqrt{5-x^{2}}}$

Question 35

Find $\lim _{x \rightarrow 1} \frac{x^{2}-3 x+2}{x^{2}-1}$

Question 36

Find $\lim _{x \rightarrow 1} \frac{x^{2}-3 x+2}{x^{2}-5 x+4}$

Question 37

Find $\lim _{x \rightarrow 5} \frac{x^{2}-9 x+20}{x^{2}-6 x+5}$

Question 38

$\lim _{x \rightarrow-2}\left(\frac{\frac{1}{x}+\frac{1}{2}}{x+2}\right)$

Question 39

$\lim _{x \rightarrow 2} \frac{3 x^{2}-x-10}{x^{2}-4}$

Question 40

$\lim _{x \rightarrow 2} \frac{x^{3}-2 x^{2}}{x^{2}-5 x+6}$

Question 41

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

Question 42

$\lim _{x \rightarrow 3} \frac{x^{4}-81}{2 x^{2}-5 x-3}$

Question 43

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

Question 44

find $\lim _{x \rightarrow 2} \frac{2 x^{3}+3 x^{2}-8 x-12}{x^{3}+5 x^{2}-4 x-20}$

Question 45

If $f(x)=\frac{1}{x}$. prove that $\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=-\frac{1}{x^{2}}$

Question 46

Find $\lim _{x \rightarrow 2}\left(\frac{1}{x-2}-\frac{4}{x^{2}-4}\right)$

Question 47

Find $\lim _{x \rightarrow 2}\left[\frac{1}{x-2}-\frac{2(2 x-3)}{x^{3}-3 x^{2}+2 x}\right]$

Question 48

$\lim _{x \rightarrow 1}\left[\frac{x-2}{x^{2}-x}-\frac{1}{x^{3}-3 x^{2}+2 x}\right]$