Rationalization Method to Remove Indeterminate Form

Q.

Evaluate $\underset{x\to \sqrt{10}}{lim}\frac{\sqrt{7-2x}-(\sqrt{5}-\sqrt{2})}{x^2-10}$

Or

Differentiate $x^{2}cosx$ by first principle.

Q.

If f(9) = 9, f'(9) = 4, then $\underset{x\to 9}{lim}\frac{\sqrt{f\left(x\right)}-3}{\sqrt{x}-3}$ equals

1. 
2. 
3. 
4. 

Q.

The value of $I={\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}|\sin x|dx$ is

1. $0$

2. $2$

3. $-2$

4. $-2<I<2$

Q.

$\underset{x\to a}{\lim }\left(\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}\right)=$

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

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

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

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

Q.

$\underset{x\to 0}{\lim }\frac{(1-e^x)\sin x}{(x^2+x^3)}$

1. $-1$

2. $0$

3. $1$

4. $2$

Q.

Evaluate: $\underset{x\to 2}{\lim }\left(\frac{\sqrt{1+\sqrt{2+x}}-\sqrt{3}}{x-2}\right)$

1. $\frac{1}{8\sqrt{3}}$

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

3. $8\sqrt{3}$

4. $\sqrt{3}$

Q.

Evaluate: $\underset{x\to 0}{\lim }\left(\frac{x}{\sqrt{1+x}-\sqrt{1-x}}\right)$

1. $0$

2. $1$

3. $2$

4. $-1$

5. $-2$

Q.

$\underset{x\to 2}{lim}\frac{\sqrt{x-2}+\sqrt{x}-\sqrt{2}}{\sqrt{x^2-4}}\text{is equal to}$

1. 1

2. 1/2

3. 2

4. -1

Q.

$(7+4\sqrt{3}{)}^{100}$ = I + f, where I is the integral part and f is the fractional part of

$(7+4\sqrt{3}{)}^{100}$. Find the value of (I + f)(1 - f)

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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. $\sqrt{\pi }$

Q.

if $\sum _{r=0}^n\left\{a_r\right(x-y+2{)}^r-b_r(y-x-1{)}^r\}=0,\forall$ n $\in$ N, then $b_n$ is equal to

1. a_n

2. (-1)ⁿa_n

3. (-1)ⁿ2ⁿa_n

4. 2ⁿa_n

Q. If f(9) = 9, f'(9) = 4, then $\underset{x\to 9}{lim}\frac{\sqrt{f\left(x\right)}-3}{\sqrt{x}-3}$ equals

1. $2$
2. $4$
3. $6$
4. $0$

Q. If G(x) = - $\sqrt{25-x^2}$ then $\underset{x\to 1}{lim}\frac{G\left(x\right)-G\left(1\right)}{x-1}$ has the value

1. $\frac{1}{24}$
2. $\frac{1}{5}$
3. $-\sqrt{24}$
4. $\frac{1}{2\sqrt{6}}$