Standard Limits to Remove Indeterminate Form

Q.

$\underset{x\to \frac{\pi }{2}}{lim}\frac{(1-tan\left(\frac{x}{2}\right))(1-sinx)}{(1+tan\left(\frac{x}{2}\right))(\pi -2x{)}^3}is$

1. 1/8

2. 0

3. ∞

4. 1/32

Q.

If $li{m}_{x\to 0}$kx cosec x = $li{m}_{x\to 0}$x cosec kx , then k =

1. 1

2. 

3. 

4. -1

Q. $\underset{x\to -\infty }{lim}\left\{\frac{x^4\sin \left(\frac{1}{x}\right)+x^2}{1+|x{|}^3}\right\}$ is equal to

1. 
2. 
3. 
4. does not exist

Q.

If 0 < a < b, then $\underset{n\to \infty }{lim}(b^n+a^n{)}^{1/n}$ is equal to

1. e

2. a

3. b

4. 1

Q. The value of $\underset{x\to 0}{lim}\underset{n\to \infty }{lim}({\sec }^{n}x{)}^{{\cos }^{n}x}$ is equal to

Q.

$\underset{x\to 0}{lim}\frac{1-cosxcos2xcos3x}{si{n}^{2}2x}$ is equal to

1. 7/2

2. 7/3

3. 7/4

4. 7/5

Q. If $\alpha$ is a repeated root of $a{x}^2+bx+c=0$ then $\underset{x\to \alpha }{lim}\frac{\sin (a{x}^2+bx+c)}{(x-\alpha {)}^2}$ is

1. 
2. 
3. 
4. 

Q.

$li{m}_{x\to 0}\frac{(e^{1/x}-1)}{(e^{1/x}+1)}$

1. 0

2. 1

3. -1

4. Does not exist

Q.

The value of ${lim}_{x\to \infty }x[ta{n}^{-1}\left(\frac{x+1}{x+2}\right)-ta{n}^{-1}\left(\frac{x}{x+2}\right)]is$

1. 1

2. -1

3. 1/2

4. -1/2

Q.

$li{m}_{x\to 0}$ $\frac{1-cos2x}{x}$
[MMR 1983]

1. 4

2. 0

3. 1

4. 2