Indeterminate Forms

Q. If $L=\underset{n\to \infty }{lim}{n}^{-n^2}{[(n+1)(n+\frac{1}{2})(n+\frac{1}{2^2})\cdots (n+\frac{1}{2^{n-1}})]}^n$, then the value of $\ln L$ is

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

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

Q.

Which of the following limit is not in the indeterminant form ?

1. 

2. 

3. 

4. 

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.

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

1. 10²

2. 10

3. 10³

4. 10⁴

Q.

$\underset{x\to \infty }{lim}{\left(\frac{3x^2+2x+1}{x^2+x+2}\right)}^{\frac{6x+1}{3x+2}}\text{is equal to}$

1. 3

2. 9

3. 1

4. 4

Q.

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

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

Q.

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

1. 1

2. e

3. 0

4. 1/e

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. 

Q.

$\underset{n\to \infty }{lim}\frac{1+\frac{1}{2}+\frac{1}{2^2}+....+\frac{1}{2^n}}{1+\frac{1}{3}+\frac{1}{3^2}+....+\frac{1}{3^n}}\text{is equal to}$

1. 2/3

2. 1/3

3. 4/3

4.