KC Sinha Mathematics Solution Class 11 Chapter 27 Limits of a Function Exercise 27.2

 Exercise 27.2

Page no 27.39

Question 1

Evaluate the following limits :

(i) $\lim _{\theta \rightarrow 0} \frac{\sin 5 \theta}{\tan 6 \theta}$

(ii) $\lim _{x \rightarrow 0} \frac{\sin 2 x}{\sin 3 x}$

(iii) $\lim _{x \rightarrow 0} \frac{\sin \alpha x}{\sin \beta x}$, where $\alpha$ and $\beta$ are fixed non-zero real numbers.

(iv) $\lim _{x \rightarrow 0} \frac{\tan 2 x}{\tan 5 x}$

(v) $\lim _{x \rightarrow 0} \frac{\sin 3 x}{\sin 7 x}$

(vi) $\lim _{x \rightarrow 0} \frac{\sin a x}{b x}$

(vii) $\lim _{x \rightarrow 0} \frac{\sin 4 x}{\sin 2 x}$

(viii) $\lim _{x \rightarrow 0} x \sec x$

(ix) $\lim _{x \rightarrow 0} \frac{\cos x}{\pi-x}$

(x) $\lim _{x \rightarrow 0} \frac{\sin a x+b x}{a x+\sin b x}$

(xi) $\lim _{x \rightarrow 0} \frac{a x+x \cos x}{b \sin x}$

Question 2

(i) Find $\lim _{\theta \rightarrow 0} \theta(3 \operatorname{cosec} 2 \theta-2 \cot 3 \theta)$ (ii) Find $\lim _{x \rightarrow 0} \frac{\sin 7 x-\sin x}{\sin 6 x}$

Question 3

$\lim _{x \rightarrow 0}(\operatorname{cosec} x-\cot x)$

Question 4

Find $\lim _{x \rightarrow 0} \frac{\cos 7 x-\cos 9 x}{\cos 3 x-\cos 5 x}$

Question 5

 Find $\lim _{x \rightarrow 0} \frac{3 \sin x-\sin 3 x}{x^{3}}$

Question 6

Find $\lim _{x \rightarrow 0} \frac{1-\cos k x}{x^{2}},(k \neq 0)$

Question 7

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

Question 8

Find (i) $\lim _{x \rightarrow 0} \frac{1-\cos 4 x}{1-\cos 5 x}$ 

(ii) $\lim _{x \rightarrow 0} \frac{\cos 2 x-1}{\cos x-1}$

Question 9

Find $\lim _{x \rightarrow 0} \frac{1-\cos m x}{1-\cos n x}$, where $m$ and $n$ are fixed non-zero real numbers.

$2 \sin x-\sin 2 x$

Question 10

Find $\lim _{x \rightarrow 0} \frac{\tan x-\sin x}{\sin ^{3} x}$

Question 11

Find $\lim _{x \rightarrow 0} \frac{2 \sin x-\sin 2 x}{x^{3}}$

Question 12

Find $\lim _{x \rightarrow 0} \frac{\sin (2+x)-\sin (2-x)}{x}$

Question 13

Find $\lim _{x \rightarrow 0} \frac{\tan x-\sin x}{x^{2}}$

Page no 27.40

Question 14

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

Question 15

Find $\lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{\sin ^{2} 2 \theta}$

Question 16

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

Question 17

Find $\lim _{x \rightarrow 0} \frac{\tan 2 x-\sin 2 x}{x^{3}}$

Question 18

Find $\lim _{x \rightarrow 0} \frac{3 \sin x-\sin 3 x}{x(\cos 2 x-\cos 4 x)}$

Question 19

Find $\lim _{x \rightarrow 0} \frac{\cos a x-\cos b x}{\cos x-1}$

Question 20

 Find $\lim _{\alpha \rightarrow 0} \frac{\sin \alpha{ }^{n}}{(\sin \alpha)^{m}}, n>m>0$

Question 21

 Find $\lim _{h \rightarrow 0} \frac{\sin (a+3 h)-3 \sin (a+2 h)+3 \sin (a+h)-\sin a}{h^{3}}$

Question 22

Find $\lim _{h \rightarrow 0} \frac{\tan (a+2 h)-2 \tan (a+h)+\tan a}{h^{2}}$

Question 23

Find (i) $\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{\pi}{2}-x\right) \tan x$

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

Question 24

Find

$\lim _{x \rightarrow \frac{\pi}{2}} \frac{1-\sin x}{\cos x}$

Question 25

Find

(i) $\lim _{x \rightarrow \frac{\pi}{2}} \frac{2 \cos x}{\pi-2 x}$

(ii) $\lim _{x \rightarrow \pi} \frac{\sin (\pi-x)}{\pi(\pi-x)}$

Question 26

Find $\lim _{x \rightarrow \frac{\pi}{2}} \frac{1-\sin x}{\left(\frac{\pi}{2}-x\right) \cot x}$

Question 27

Find $\lim _{x \rightarrow 1}(1-x) \tan \frac{\pi x}{2}$

Question 28

Find $\lim _{x \rightarrow \pi} \frac{1+\cos x}{\tan ^{2} x}$

Question 29

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

Question 30

Find $\lim _{\theta \rightarrow \frac{\pi}{4}} \frac{1-\tan \theta}{1-\sqrt{2} \sin \theta}$

Question 31

Find $\lim _{y \rightarrow a} \sin \frac{y-a}{2} \tan \frac{\pi y}{2 a}$

Question 32

Find $\lim _{\theta \rightarrow \frac{\pi}{2}} \frac{\sec \theta-\tan \theta}{\pi-2 \theta}$

Question 33

Find $\lim _{\theta \rightarrow \frac{\pi}{2}} \frac{\cot \theta-\cos \theta}{\cos ^{3} \theta}$

Question 34

Find $\lim _{x \rightarrow 1}(1+\cos \pi x) \cot ^{2} \pi x$

Question 35

Find $\lim _{x \rightarrow y} \frac{\sin x-\sin y}{x-y}$

Question 36

Find $\lim _{a \rightarrow \frac{\pi}{4}} \frac{\sin \alpha-\cos \alpha}{\alpha-\frac{\pi}{4}}$

Question 37

Find

$\lim _{y \rightarrow \theta} \frac{y \cos \theta-\theta \cos y}{y-\theta}$

Question 38

Find $\lim _{a \rightarrow a} \frac{\sin x-\sin a}{\sqrt{x}-\sqrt{a}}$

Question 39

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