Indeterminate Forms

Q.

The value of $\underset{x\to \infty }{\lim }{\left[\frac{x^2-2x+1}{x^2-4x+2}\right]}^x$ is

1. $e^2$

2. $e^{-2}$

3. $e^6$

4. None of these

Q. If $\underset{x\to 0}{lim}(1+a\sin x{)}^{cosecx}=3,$ then a is

1. In 3
2. In 2
3. In 4
4. 

Q.

$\text{The limiting value of}{\left(cosx\right)}^{1/sinx}\text{as~x}\to 0is$

1. 0

2. 1/e

3. 1

4. e

Q. $\underset{n\to \infty }{lim}\frac{{2.3}^n-{3.5}^n}{{3.3}^n+{4.5}^n}=$

1. 
2. 
3. 
4. 

Q.

$\text{The value of}\underset{n\to \infty }{lim}\frac{\sqrt[4]{4n^5+2}-\sqrt[3]{3n^2+1}}{\sqrt[5]{5n^4+2}-\sqrt[2]{2n^3+1}}is$

1. 1

2. 0

3. ∞

4. -1

Q.

$li{m}_{x\to 1}$ $\frac{logx}{x-1}$ =
[Rpet 1996; MP PET 1996; P.CET 2002]

1. 

2. 1

3. -1

4. 0

Q.

$\underset{x\to \infty }{lim}\frac{lo{g}_{\epsilon }\left[x\right]}{x},\text{where[x]denotes the greatest integer less than or equal to x, is:}$

1. 1

2. -1

3. 0

4. Does not exist

Q.

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

Q.

$\underset{x\to 1}{lim}\frac{logx}{x-1}$

___

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

1. $-1$
2. $1$
3. $0$
4. $\infty$

Q.

$\underset{x\to \infty }{lim}\frac{\sqrt{x^2+1}-\sqrt[3]{3x^2+1}}{\sqrt[4]{4x^4+1}-\sqrt[5]{5x^4-1}}isequalto$

1. 1

2. -1

3. 0

4. 1/2

Q. $\underset{x\to 1}{lim}\frac{logx}{x-1}=$

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

Q. 21. lim (cosec x -cotx)x-30

Q.

$\underset{x\to \infty }{lim}{\left(\frac{x^5+5x+3}{x^2+x+2}\right)}^x\text{equals}$

1. e³

2. e⁴

3. e²

4. e

Q.

$\underset{x\to \infty }{lim}\frac{\sqrt{x^2+1}-\sqrt[3]{3x^2+1}}{\sqrt[4]{4x^4+1}-\sqrt[5]{5x^4-1}}isequalto$

1. 1

2. -1

3. 0

4. 1/2

Q. Evaluate the Given limit: $\underset{x\to 0}{lim}(\cos ecx-\cot x)$

Q. $\underset{n\to \infty }{lim}\frac{a^n+b^n}{a^n-b^n}$, where $a>b>1$, is equal to

1. $-1$
2. $1$
3. $0$
4. $\frac{a}{b}$

Q.

Find $\underset{x\to 0}{\text{lim}}$ f(x) and $\underset{x\to 1}{\text{lim}}$ f(x) where f(x)= $\{\begin{array}{cc}2x+3& x\le 0\\ 3(x+1)& x>0\end{array}$

Q. Let $f$ be a biquadratic function of $x$ given by $f\left(x\right)=A{x}^4+B{x}^3+C{x}^2+Dx+E,$ where $A,B,C,D,E\in R$ and $A\ne 0.$ If $\underset{x\to 0}{lim}{\left(\frac{f(-x)}{2x^3}\right)}^{1/x}=e^{-3},$ then

1. $A+4B=0$
2. $A-3B=0$
3. $f\left(1\right)=8$
4. $f^{\prime }\left(1\right)=-30$

Q. $\underset{x\to \infty }{lim}\frac{(2+x{)}^{40}(4+x{)}^5}{(2-x{)}^{45}}$

1. $-1$
2. $1$
3. $16$
4. $32$

Q.

k = $\underset{x\to \infty }{lim}\left(\frac{\sum _{k=1}^{1000}(x+k{)}^m}{x^m+{10}^{1000}}\right)$ is (m > 101)

1. 10

2. ${10}^2$

3. ${10}^3$

4. ${10}^4$

Q. $\underset{n\to \infty }{lim}\frac{{2.3}^n-{3.5}^n}{{3.3}^n+{4.5}^n}=$

1. $\frac{2}{3}$
2. $-\frac{3}{4}$
3. $1$
4. $0$

Q. $\underset{x\to 0}{lim}{\left(\frac{3x^2+2}{7x^2+2}\right)}^{1/x^2}$ is equal to:

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

Q.

$\text{The value of}\underset{n\to \infty }{lim}\frac{\sqrt[4]{4n^5+2}-\sqrt[3]{3n^2+1}}{\sqrt[5]{5n^4+2}-\sqrt[2]{2n^3+1}}is$

1. 1

2. 0

3. -1

4. $\infty$

Q. The value of $\underset{x\to 1}{lim}{\left(\tan \frac{x\pi }{4}\right)}^{\tan \frac{\pi x}{2}}$ is

1. 
2. 
3. 
4. 

Q.

If both $\underset{x\to a}{lim}f\left(x\right)$ and $\underset{x\to a}{lim}g\left(x\right)$ and exist finitely and $\underset{x\to a}{lim}g\left(x\right)=0$, then
$\underset{x\to a}{lim}\frac{f\left(x\right)}{g\left(x\right)}=\frac{\underset{x\to a}{lim}f\left(x\right)}{\underset{x\to a}{lim}g\left(x\right)}$

1. True

2. False

Q. The value of $\underset{n\to \infty }{lim}\frac{4\cdot 5^{n+1}-2\cdot 3^n}{6\cdot 3^{n+1}+5^n}$ is

1. $-20$
2. $20$
3. $1$
4. does not exist

Q.

$\text{The value of}\underset{x\to \infty }{lim}[\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}]\text{is}.$

1. 1/2

2. 1

3. 0

4. -1/2

Q.

$\underset{x\to 1}{lim}\frac{logx}{x-1}$

Q.

$li{m}_{n\to \infty }$ $\frac{n\left(\right(2n+1{)}^2)}{(n+2)(n^2+3n-1)}$ =

1. 0

2. 2

3. 4

4. ∞