Rationalization Method to Remove Indeterminate Form

Q. The value of $\underset{x\to 2}{lim}\frac{2^x+2^{3-x}-6}{\sqrt{2^{-x}}-2^{1-x}}$ is

1. $16$
2. $8$
3. $4$
4. $2$

Q. If $f\left(x\right)=9^{-2/x^2},x\ne 0$ is continuous at $x=0,$ then the value of $f\left(0\right)$ is

1. $-1$
2. $1$
3. $9$
4. $0$

Q.

$li{m}_{x\to a}$$\frac{\left(\sqrt{3x-a}\right)-\left(\sqrt{x+a}\right)}{x-a}$ =

1. $\sqrt{2a}$

2. $\frac{1}{\sqrt{2a}}$

3. 2a

4. $\frac{1}{2a}$

Q. If $\alpha$ is positive root of the equation, $p\left(x\right)=x^2-x-2=0,$, then $\underset{x\to {\alpha }^{+}}{lim}\frac{\sqrt{1-\cos \left(p\left(x\right)\right)}}{x+\alpha -4}$ is equal to :

1. $\frac{1}{\sqrt{2}}$
2. $\frac{1}{2}$
3. $\frac{3}{2}$
4. $\frac{3}{\sqrt{2}}$

Q.

$li{m}_{x\to 1}$$\frac{(2x-3)(\sqrt{x}-1)}{({2x}^2+x-3)}$ =
[IIT 1977]

1. -1/10

2. 1/10

3. -1/8

4. None of these

Q.

$li{m}_{x\to a}$$\frac{\left(\sqrt{3x-a}\right)-\left(\sqrt{x+a}\right)}{x-a}$ =

1. $\sqrt{2a}$

2. $\frac{1}{\sqrt{2a}}$

3. 2a

4. $\frac{1}{2a}$

Q. $\underset{x\to 1^{-}}{lim}\frac{\sqrt{\pi }-\sqrt{2{\sin }^{-1}x}}{\sqrt{1-x}}$ is equal to :

1. $\sqrt{\frac{2}{\pi }}$
2. $\sqrt{\frac{\pi }{2}}$
3. $\frac{1}{\sqrt{2\pi }}$
4. $\sqrt{\pi }$