Standard Limits to Remove Indeterminate Form

Q. The value of the limit $\underset{x\to e}{\lim }\frac{\log x-1}{x-e}$ is

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

Q.

If ${lim}_{x\to 3}\frac{x^n-3^n}{x-3}=108$, find the value of n.

Q.

If ${lim}_{x\to a}\frac{x^9-a^9}{x-a}=9$, find all possible values of a.

Q.

${lim}_{x\to 1}\frac{x^{15}-1}{x^{10}-1}$

Q.

If ${lim}_{x\to 0}kxcosecx={lim}_{x\to 0}xcoseckx$, find k.

Q. Let $\left[x\right]$ denote the greatest integer less than or equal to $x$. Then :
$\underset{x\to 0}{lim}\frac{\tan \left(\pi {\sin }^{2}x\right)+(|x|-\sin (x\left[x\right]){)}^2}{x^2}$:

1. equals $\pi +1$
2. equals $\pi$
3. equals $0$
4. does not exist

Q. The value of $\underset{h\to 0}{lim}2\left\{\frac{\sqrt{3}\sin (\frac{\pi }{6}+h)-\cos (\frac{\pi }{6}+h)}{\sqrt{3}h(\sqrt{3}\cos h-\sin h)}\right\}$ is :

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

Q.

${lim}_{x\to 0}\frac{2sinx-sin2x}{x^3}$

Q. If $\alpha$ and $\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0,$ then $\underset{x\to \alpha }{lim}\frac{1-\cos (a{x}^2+bx+c)}{(x-\alpha {)}^2}$ is :

1. $\frac{(\alpha -\beta {)}^2}{2}$
2. $2(\alpha -\beta {)}^2$
3. $2a^2(\alpha -\beta {)}^2$
4. $\frac{a^2(\alpha -\beta {)}^2}{2}$

Q.

${lim}_{h\to 0}[\frac{1}{h3\sqrt{8+h}}-\frac{1}{2h}]=$

1. $\frac{-1}{12}$
   

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

3. $\frac{-16}{3}$
   

4. $\frac{-1}{48}$
   

Q. $\underset{x\to 0}{lim}\frac{{\sin }^{2}x}{\sqrt{2}-\sqrt{1+\cos x}}$ equals:

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

Q. For each $t\in R$, let $\left[t\right]$ be the greatest integer less than or equal to $t$. Then, $\underset{x\to 1^{+}}{lim}\frac{(1-|x|+\sin |1-x|)\sin \left(\frac{\pi }{2}\right[1-x\left]\right)}{|1-x|[1-x]}$

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

Q.

${lim}_{x\to 0}\frac{3sinx-sin3x}{x^3}$

Q.

${lim}_{x\to a}{\left\{\frac{\sin x}{\sin a}\right\}}^{\frac{1}{x-a}}$

Q.

If ${lim}_{x\to a}\frac{x^5-a^5}{x-a}=405$, find all possible values of a.

Q.

${lim}_{x\to a}\frac{logx-loga}{x-a}$

Q. The value of $\underset{x\to 0}{lim}\frac{|x|}{x}$ is

1. -1
2. 0
3. Does not exist
4. 1

Q.

The modulus of $\frac{\left(1-i\right)}{(3+i)}+\frac{4i}{5}$ is

1. $\sqrt{5}$ unit

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

3. $\frac{\sqrt{11}}{5}$ unit

4. $\frac{\sqrt{12}}{5}$ unit

Q.

$\underset{x\to \infty }{lim}\left[\frac{\sum n^2}{n^3}\right]$

1. 

2. 

3. 

4. 

Q.

${lim}_{x\to a}\frac{cos\sqrt{x}-cos\sqrt{a}}{x-a}$

Q.

${lim}_{x\to 4}\frac{x^3-64}{x^2-16}$

Q.

${lim}_{x\to 0}\frac{(1+x{)}^6-1}{(1+x{)}^2-1}$

Q.

Evaluate ${lim}_{x\to 0}\frac{7xcosx-3sinx}{4x+tanx}$

Q. $\underset{x\to 0}{lim}\frac{(1-\cos 2x{)}^2}{2x\tan x-x\tan 2x}$ is:

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

Q.

${lim}_{x\to 0}\frac{cosax-cosbx}{coscx-1}$

Q.

Evaluate ${lim}_{x\to 0}\frac{\log (a+x)-\log (a-x)}{x}$

Q.

${lim}_{x\to 27}\frac{(x^{\frac{1}{3}}+3)(x^{\frac{1}{3}}-3)}{x-27}$

Q.

${lim}_{x\to \frac{\pi }{4}}\frac{1-sin2x}{1+cos4x}$

Q. If $\underset{x\to 1}{lim}\frac{x^4-1}{x-1}=\underset{x\to k}{lim}\frac{x^3-k^3}{x^2-k^2}$ , then $k$ is :

1. $\frac{4}{3}$
2. $\frac{8}{3}$
3. $\frac{3}{2}$
4. $\frac{3}{8}$

Q.

${lim}_{x\to 0}\frac{2x-sinx}{tanx+x}$