### Mathematics Assignments | PDF | 8 to 12

PDF Files of Mathematics Assignments From VIII Standard to XII Standard PDF of mathematics Assignments for the students from VIII standard to XII standard.These assignments are strictly according to the CBSE and DAV Board Final question Papers

## Math Assignment Class XI | Chapter 12 | Limits

Math Assignment Class XI Chapter 12 Limits and derivatives, Extra questions on Limits strictly according to the CBSE & DAV Board, Important questions on limits.

## Math Assignment Class XI | Chapter 12 | Limits

Question 1
Evaluate the  limits: $\displaystyle \lim_{x \to 2}\frac{\frac{1}{x}-\frac{1}{2}}{x-2}$
Ans: -1/4
Question 2 :
Evaluate the  limits:  $\displaystyle \lim_{x \to 0}\frac{\sqrt{1-x}-1}{x}$
Ans: - 1/2

Question 3:
Evaluate the  limits   $\lim_{x\rightarrow 0}\frac{cot2x-cosec2x}{x}$
Ans:  -1

Question 4 :
Evaluate the  limits $\lim_{x\rightarrow 0}\frac{sinx-2sin3x+sin5x}{x}$
Ans: 0

Question 5:
Evaluate the  limits  $\lim_{x\rightarrow 0}\; \frac{sin2x+sin6x}{sin5x-sin3x}$
Ans: 4

Question 6 :
Evaluate the  limits $\lim_{x\rightarrow 0}\;\; \frac{sin^{2}3x}{x^{2}}$
Ans: 9

Question 7 :

If f(x) = $\left\{\begin{matrix}3x+2, x\leq 0 \\\\\; \; 2(x+1), x> 0\end{matrix}\right.$        then find  $\displaystyle \lim_{x \to 0}f(x)$

Ans: 2
Solution Hint:  Find LHL and RHL

Question 8:
Evaluate the  limits: $\displaystyle \lim_{x \to 0}\frac{Log(1+3x)}{x}$
Ans: 3
Solution Hint:
Use formula:  $\displaystyle \lim_{x \to 0}\frac{Log(1+x)}{x}=1$

Question 9 :

If  $\displaystyle \lim_{ x\to 0}\frac{sin3x}{sin7x} = K$  , then find the value of K

Ans:  3/7

Question 10 :
Evaluate the  limits: $\lim_{x\rightarrow 9 }\;\; \frac{x^{3/2}-27}{x-9}$
Ans: 9/2

Question 11 :

If  $\displaystyle \lim_{x \to 1}\frac{x^{4}-1}{x-1}=\displaystyle \lim_{x \to k}\frac{x^{3}-k^{3}}{x^{2}-k^{2}}$   then find the value of k

Ans: 8/3

Question 12 :
Evaluate the  limits: $\lim_{x\rightarrow 4 }\;\; \frac{x^{2}-16}{\sqrt{x^{2}+9}-5}$
Ans: 10

Question 13 :
Evaluate the  limits: $\lim_{x\rightarrow 2 }\;\; \frac{x^{2}-4}{\sqrt{3x-2}-\sqrt{x+2}}$
Ans: 8

Question 14 :
Evaluate the  limits: $\lim_{x\rightarrow 4 }\;\; \frac{(x^{2}-x-12)^{18}}{(x^{3}-8x^{2}+16x)^{9}}$
Ans: $Ans:\left ( \frac{7}{2} \right )^{18}$

Question 15 :
Evaluate the  limits: $\lim_{x\rightarrow 1 }\;\; \left [ \frac{2}{1-x^{2}}+\frac{1}{x-1} \right ]$
Ans: 1/2

Question 16 :
Evaluate the  limits: $\lim_{x\rightarrow 2 }\;\; \frac{x^{3}-6x^{2}+11x-6}{x^{2}-6x+8}$
Ans: 1/2

Question 17 :
Evaluate the  limits: $\lim_{x\rightarrow 2 }\;\; \frac{x^{3}-3x^{2}+4}{x^{4}-8x^{2}+16}$
Ans: 3/16

Question 18 :
Evaluate the  limits $\lim_{x\rightarrow 1/2 }\;\; \frac{8x^{3}-1}{16x^{4}-1}$
Ans: 3/4

Question 19 :
Evaluate the  limits      $\lim_{x\rightarrow 0}\left [ \frac{\sqrt{1+tanx}-\sqrt{1-tanx}}{sin2x} \right ]$

Ans: 1/2

Solution Hint:

Rationalizing the numerator.

Simplify and then putting the limit.

Question 20 $\lim_{x\rightarrow \frac{\pi }{3}}\frac{\sqrt{3}-tanx}{\pi -3x}$
Ans: 4/3

Question 21
Evaluate the  limits:  $\lim_{x\rightarrow \frac{\pi }{2}}\frac{1-sinx}{cos^{2}x}$
Ans: 1/2

Question 22
Evaluate the  limits:  $\lim_{x\rightarrow \pi }\frac{sin3x-3sinx}{(\pi -x)^{3}}$
Ans: - 4

Solution Hint

Use  Sin3x = 3sinx – 4sin3x,

putting  x = Ï€+h, as x→ Ï€, then h→0

Question 23
Evaluate the  limits: $\displaystyle \lim_{x \to 0}\frac{1-cos2x}{x(e^{5x}-1)}$

Ans: 2/5
Solution Hint:   $\displaystyle \lim_{x \to 0}\frac{1-cos2x}{x(e^{5x}-1)}$    = $\displaystyle \lim_{x \to 0}\frac{2sin^{2}x}{5x(\frac{e^{5x}-1}{5})}$

= $\displaystyle \lim_{x \to 0}\frac{\frac{2sin^{2}x}{x^{2}}}{5(\frac{e^{5x}-1}{5x})}$    = 2/5

Question 24
Evaluate the  limits:  $\displaystyle \lim_{x \to 3}\frac{x^{2}-5x+6}{x-3}$
Ans: 1

Question 25
Evaluate the  limits:  $\displaystyle \lim_{x \to \frac{\pi }{4}}\frac{cosx-sinx}{x-\frac{\pi }{4}}$
Ans: - √2

Solution Hint:
$\displaystyle \lim_{x \to \frac{\pi }{4}}\frac{cosx-sinx}{x-\frac{\pi }{4}}$ = $\displaystyle \lim_{x \to \frac{\pi }{4}}\frac{\sqrt{2}\left ( \frac{1}{\sqrt{2}}cosx-\frac{1}{\sqrt{2}}sinx \right )}{x-\frac{\pi }{4}}$

= $\displaystyle \lim_{x \to \frac{\pi }{4}}\frac{\sqrt{2}\left (sin \frac{\pi }{4}cosx-cos\frac{\pi }{4}sinx \right )}{x-\frac{\pi }{4}}$

=  $\displaystyle \lim_{x \to \frac{\pi }{4}}\frac{\sqrt{2}sin\left ( \frac{\pi }{4}-x \right )}{x-\frac{\pi }{4}}$

=  $\displaystyle \lim_{x \to \frac{\pi }{4}}\frac{-\sqrt{2}sin\left (x- \frac{\pi }{4} \right )}{x-\frac{\pi }{4}}$

=  $-\sqrt{2}\times 1$ =  $-\sqrt{2}$

Question 26
Evaluate the  limits:   $\displaystyle \lim_{x \to \frac{\pi }{6}}\left [ \frac{\sqrt{3}sinx-cosx}{x-\frac{\pi }{6}} \right ]$
Ans: 2
Solution Hint:

=$\displaystyle \lim_{x \to \frac{\pi }{6}}\left [ \frac{2\left ( \frac{\sqrt{3}}{2}sinx-\frac{1}{2}cosx \right )}{x-\frac{\pi }{6}} \right ]$

=$\displaystyle \lim_{x \to \frac{\pi }{6}}\left [ \frac{2\left ( cos\frac{{\pi }}{6}sinx-sin\frac{\pi }{6}cosx \right )}{x-\frac{\pi }{6}} \right ]$

=$\displaystyle \lim_{x \to \frac{\pi }{6}}2\left [ \frac{sin\left (x-\frac{{\pi }}{6} \right )}{x-\frac{\pi }{6}} \right ]$

=2 x 1 = 2

Question 27
Evaluate the  limits: $\displaystyle \lim_{x \to 0}\frac{10^{x}-5^{x}-2^{x}+1}{x^{2}}$
Ans: Log5.Log2
Solution Hint:

=  $\displaystyle \lim_{x \to 0}\frac{2^{x}\times 5^{x}-5^{x}-2^{x}+1}{x^{2}}$

=  $\displaystyle \lim_{x \to 0}\frac{5^{x}( 2^{x}-1)-1(2^{x}-1)}{x^{2}}$

=  $\displaystyle \lim_{x \to 0}\frac{(5^{x}-1)( 2^{x}-1)}{x^{2}}$

=  $\displaystyle \lim_{x \to 0}\left ( \frac{5^{x}-1}{x} \right )\displaystyle \lim_{x \to 0}\left ( \frac{2^{x}-1}{x} \right )$

=  Log5.Log2
Formula used :  $\displaystyle \lim_{x \to 0}\frac{e^{x}-1}{x}=Loge=1$

Question 28
Evaluate the  limits:   $\displaystyle \lim_{x \to 0}\frac{cos2x-cos3x}{cos4x-1}$
Ans: -5/16
Solution Hint:

$=\displaystyle \lim_{x \to 0}\frac{-(cos3x-cos2x)}{-(1-cos4x)}$

= $\displaystyle \lim_{x \to 0}\frac{-2sin\frac{5x}{2}sin\frac{x}{2}}{2sin^{2}2x}$

= $-\displaystyle \lim_{x \to 0}\frac{sin\frac{5x}{2}}{x}\times \frac{sin\frac{x}{2}}{x}\div \left ( \frac{sin2x}{x} \right )^{2}$

= $-\displaystyle \lim_{x \to 0}\left ( \frac{sin\frac{5x}{2}}{\frac{5x}{2}}\times \frac{5}{2} \right ) \left ( \frac{sin\frac{x}{2}}{\frac{x}{2}}\times \frac{1}{2} \right )\div \left ( \frac{sin2x}{2x} \times 2\right )^{2}$

= $-\left ( \frac{5}{2}\times \frac{1}{2}\div 2^{2} \right )$    = $-\frac{5}{16}$

Question 29
Evaluate the  limits:

$\displaystyle \lim_{x \to 0}\frac{cos^{2}x-sin^{2}x-1}{\sqrt{x^{2}+1}-1}$
Ans: - 4

Solution Hint

$\displaystyle \lim_{x \to 0}\frac{\left ( cos^{2}x-sin^{2}x \right )-1}{\sqrt{x^{2}+1}-1}$

$\displaystyle \lim_{x \to 0}\frac{cos2x-1}{\sqrt{x^{2}+1}-1}$

$\displaystyle \lim_{x \to 0}\frac{1-cos2x}{1-\sqrt{x^{2}+1}}$

$\displaystyle \lim_{x \to 0}\frac{2sin^{2}x}{1-\sqrt{x^{2}+1}}$

$\displaystyle \lim_{x \to 0}\frac{2sin^{2}x}{1-\sqrt{x^{2}+1}}\times \frac{1+\sqrt{x^{2}+1}}{1+\sqrt{x^{2}+1}}$

$\displaystyle \lim_{x \to 0}\frac{2sin^{2}x(1+\sqrt{x^{2}+1})}{1-x^{2}-1}$

$-2\displaystyle \lim_{x \to 0}\frac{sin^{2}x}{x^{2}}\times (1+\sqrt{x^{2}+1})$

$-2\times 1^{2}\times 2 =-4$

Question 30
Evaluate the  limits:

$\displaystyle \lim_{x \to 0}\left ( \frac{e^{x}+sinx-1}{3x} \right )$
Ans: 2/3
Question 31
Find n if   $\displaystyle\lim_{x\to 2}\frac{x^{n}-2^{n}}{x-2}=80,\:x\varepsilon N$
Question 32
Evaluate:   $\displaystyle\lim_{x\to\pi/2}\frac{1-sinx}{cosx}$