Expansions to Remove Indeterminate Form

Q.

Evaluate $\underset{x\to \infty }{\lim }x\sin \left(\frac{2}{x}\right)$

1. $\infty$

2. $0$

3. $2$

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

Q.

In $∆ABC,(a+b+c)\left(\tan \left(\frac{A}{2}\right)+\tan \left(\frac{B}{2}\right)\right)$ is equal to

1. $2ccot\left(\frac{C}{2}\right)$

2. $2acot\left(\frac{A}{2}\right)$

3. $\tan \left(\frac{C}{2}\right)$

4. None of these

Q.

The value of $\frac{\cos \left(30°\right)+i\sin \left(30°\right)}{\cos \left(30°\right)-i\sin \left(30°\right)}$is equal to

1. $i$

2. $-i$

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

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

5. $1+i$

Q.

What is the derivative of ${\sin }^{-1}\left(\frac{2x}{1+x^2}\right)$ with respect to ${\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)$.

1. $1$

2. $2$

3. $-1$

4. $4$

Q.

If ${lim}_{x\to 1}\frac{x+x^2+x^3\cdots x^n-n}{x-1}=5050,$ then n equal

1. 10

2. 100

3. 150

4. None of these

Q. Let $\alpha ,\beta \in R$ be such that $\underset{x\to 0}{lim}\frac{x^2\sin \left(\beta x\right)}{\alpha x-\sin x}=1.$ Then $6(\alpha +\beta )$ equals

Q. $\underset{x\to 0}{lim}\frac{\cos \left(\tan x\right)-\cos x}{x^4}$ is equal to

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

Q.

Find the limit: $\underset{x\to 0}{\lim }\frac{\sin \left(\left|x\right|\right)}{x}$.

Q.

If ${\sin }^{-1}x+co{t}^{-1}\left(\frac{1}{2}\right)=\frac{\mathrm{\pi }}{2}$ then $x$ is:

1. $0$

2. $\frac{1}{\sqrt{5}}$

3. $\frac{2}{\sqrt{5}}$

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

Q. The value of $\underset{x\to 1}{lim}x\text{sgn}(x-1)$ is
(where 'sgn' represents signum function)

1. $-1$
2. $1$
3. $0$
4. limit does not exist

Q. Evaluate the given limit :
$\underset{x\to 0}{lim}\frac{\cos x}{\pi -x}$

Q. The value of $\underset{x\to 0}{lim}\frac{\sqrt[3]{31+\sin x}-\sqrt[3]{31-\sin x}}{x}$ is

1. ​$\frac{2}{3}$​​​​​​
2. same as the value of $\underset{x\to 0}{lim}\frac{\sin 2x}{\tan 3x}$
3. same as the value of $\underset{x\to 0}{lim}\frac{\tan 3x}{\sin 2x}$
4. ​$\frac{3}{2}$​​​​​​

Q.

${\sin }^{-1}\left(\frac{-\sqrt{2}}{2}\right)+{\cos }^{-1}\left(\frac{-1}{2}\right)-{\tan }^{-1}\left(-\sqrt{3}\right)+{\cot }^{-1}\left(\frac{-1}{\sqrt{3}}\right)=$

1. $\frac{7\pi }{12}$

2. $\frac{17\pi }{12}$

3. $\frac{5\pi }{12}$

4. $\frac{\pi }{12}$

Q.

In $∆ABC,a=2,b=3$, and $\sin A=\frac{2}{3}$ , then $B$ is equal to

1. 

2. 

3. 

4. 

Q. Let $f\left(x\right)=1+e^{\ln \left(\ln x\right)}\cdot \ln (k^2+25)$ and $g\left(x\right)=\frac{1}{|x|-1}$. If $\underset{x\to 1}{lim}f(x{)}^{g\left(x\right)}=k(2{\sin }^2\alpha +3\cos \beta +5)$, $k>0$ and $\alpha ,\beta \in R$, then which of the following statement(s) is (are) CORRECT?

1. $k=5$
2. $\frac{{\sin }^{10}\alpha +{\cos }^5\beta }{{\sin }^2\alpha +\cos \beta }=1$
3. ${\cos }^2\beta +{\sin }^4\alpha =2$
4. ${\sin }^2\alpha >\cos \beta$

Q. The value of $\underset{x\to 0}{lim}\frac{e^{1/x}-1}{e^{1/x}+1}$ is

1. $1$
2. $-1$
3. $0$
4. It does not exist

Q.

${lim}_{x\to 4}\frac{x^2-16}{\sqrt{x}-2}$

Q.

The number of real values of $a$ satisfying the equation $a^2-2a\sin \left(x\right)+1=0$ is

1. $0$

2. $1$

3. $2$

4. Infinite

Q.

If $y={\sin }^{-1}\sqrt{\left(1-x\right)}$, then $\frac{dy}{dx}=$

1. $\frac{1}{\sqrt{\left(1-x\right)}}$

2. $\frac{-1}{2\sqrt{\left(1-x\right)}}$

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

4. $\frac{-1}{2\sqrt{x}\sqrt{\left(1-x\right)}}$

Q. Evaluate :
$i.$ $\underset{x\to 0}{lim}\frac{\sin 4x}{\sin 2x}$
$ii.$ $\underset{x\to 0}{lim}\frac{\tan x}{x}$

Q.

$\int \frac{1}{\sqrt{7-x^2}}dx=$

1. $\frac{1}{2\sqrt{7}}\log \left(\frac{\sqrt{7}+x}{\sqrt{7}-x}\right)+C$

2. ${\sin }^{-1}\left(\frac{x}{\sqrt{7}}\right)+C$

3. $\log \left|x+{\sqrt{x}}^2-7\right|+C$

4. $\frac{1}{2\sqrt{7}}\log \left|\frac{x-\sqrt{7}}{x+\sqrt{7}}\right|+C$

Q. $\underset{x\to 7/2}{lim}(2x^2-9x+8{)}^{\cot (2x-7)}$ is equal to

1. $e^{\frac{5}{2}}$
2. $e^{-\frac{5}{2}}$
3. $e^{\frac{7}{2}}$
4. $e^{\frac{3}{2}}$

Q. If $x>0$ and g is bounded function, then $\underset{n\to \infty }{lim}\frac{f\left(x\right)e^{nx}+g\left(x\right)}{e^{nx}+1}$ is

1. g(x)
2. 0
3. None of these
4. f(x)

Q.

$\underset{x\to 0}{lim}\left[\frac{lncosx}{\sqrt[4]{41+x^2}-1}\right]\text{is equal to}$

1. 2

2. -2

3. 1

4. -1

Q. If the value of $\underset{x\to 0}{lim}\frac{{\tan }^{3}x-{\sin }^{3}x}{x^5}$ is $\alpha$, then $2\alpha =$

Q. The value of $\underset{x\to 0}{lim}\frac{2x\sin 2x+3\tan x+x^2\sin x}{\sin 3x}$ is

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

Q.

${lim}_{x\to \frac{1}{4}}\frac{4x-1}{2\sqrt{x}-1}$

Q.

You are given
$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}......$;
$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}......$;
$\tan x=x+\frac{x^3}{3}+\frac{2x^5}{15}......$
Then the value of
$\underset{x\to 0}{lim}\frac{x\cos x+\sin x}{x^2+\tan x}$ is

___

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

1. $3$
2. $0$
3. $1$
4. does not exist

Q. The value of $\underset{x\to 0}{lim}\frac{1-8^x-9^x+(72{)}^x}{(9^x-8^x)(\tan 9x-\sin 8x)}$ is

1. $\log 8\cdot \log 9\cdot \log \left(\frac{9}{8}\right)$
2. $\log 8\cdot \log 9\cdot \log \left(\frac{8}{9}\right)$
3. $6\log 8\cdot \log 9\cdot \log \left(\frac{8}{9}\right)$
4. $\frac{\log 8\cdot \log 9}{\log \left(\frac{9}{8}\right)}$