Standard Limits to Remove Indeterminate Form

Q.

Find the value of $\underset{n\to \infty }{lim}\frac{1+2+3+....n}{n^2}$

1. 2

2. 

3. 

4. 1

Q.

Find the value of $\underset{x\to a}{lim}\frac{x^n-a^n}{x-a}$

1. 

2. 

3. 

4. nx

Q.

If ${\sin }^{-1}x+{\sin }^{-1}y=\frac{\pi }{2}$, then the value of ${\cos }^{-1}x+{\cos }^{-1}y=$

1. $\frac{\mathrm{\pi }}{2}$

2. $\frac{\mathrm{\pi }}{4}$

3. $\mathrm{\pi }$

4. $\frac{3\mathrm{\pi }}{4}$

Q.

$\text{The value of}{lim}_{h\to 0}\frac{In(1+2h)-2In(1+h)}{h^2}is$

1. 1

2. 0

3. -1

4. 2

Q. $\underset{x\to 0}{lim}\frac{\sin x-x+x^3}{6}$ is equal to

1. 
2. 
3. 
4. 

Q.

The value of expression $\frac{1-4sin{10}^{o}sin{70}^o}{2sin{10}^o}$ is

1. 1/2

2. 1

3. 2

4. 3

Q.

Evaluate the following limit:
$\underset{x\to -2}{\text{lim}}\frac{\frac{1}{x}+\frac{1}{2}}{x+2}$

Q.

If $li{m}_{x\to 0}$kx cosec x = $li{m}_{x\to 0}$x cosec kx , then k =

1. 1

2. -1

3. 

4. 

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. $\underset{n\to \infty }{lim}n.\cos \left(\frac{\pi }{4n}\right).\sin \left(\frac{\pi }{4n}\right)$ is equal to

1. $\frac{\pi }{4}$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{9}$
4. $\frac{\pi }{3}$

Q.

If n is an even natural number , then $\sum _{r=0}^n\frac{(-1{)}^r}{{}^n{C}_r}equals:$

1. 

2. 

3. 

4. 

Q. $\underset{x\to 0}{lim}\frac{2\sin x-\sin 2x}{x^3}$ is equal to

1. 
2. 
3. 
4. does not exist

Q. limx→a [f(x)+g(x)]=10 and limx→a f(x)=2. Then, limx→ag(x), if it exists, is

1. 2
2. 4
3. 6
4. 8

Q. 21. If nC(r-1) = 36 and nCr = 84 and nC(r+1) = 126 then value of r is : (A)9 (B)3 (C)4 (D)5 (E)6

Q. $\underset{x\to -\infty }{lim}\left\{\frac{x^4\sin \left(\frac{1}{x}\right)+x^2}{1+|x{|}^3}\right\}$ is equal to

1. 
2. does not exist
3. 
4. 

Q.

The value of $\underset{x\to a}{lim}\left(\right[\frac{100x}{sinx}]+[\frac{99sinx}{x}\left]\right)$, where [.] denotes the greatest integer function, is

1. 197

2. 198

3. 199

4. Does not exist

Q. $\underset{n\to \infty }{\lim }{\left(1+\frac{x}{n}\right)}^n$

Q. $\underset{\alpha \to \beta }{lim}\frac{{\sin }^2\alpha -{\sin }^2\beta }{{\alpha }^2-{\beta }^2}$ is equal to

1. 
2. 
3. 
4. 

Q.

If f(x)=$$\{\begin{array}{cc}x,& \text{when 0≤ x ≥ 1}\\ 2-x,& \text{2-x when 1 ≤ x ≥ 2}\end{array}$$
then $li{m}_{x\to 1}$ f(x) =

1. 2

2. 0

3. Does not exist

4. 1

Q. $\underset{x\to \infty }{lim}\frac{1+2+3+....n}{n^2}=$

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

Q. The value of $\underset{n\to \infty }{\lim }\frac{\left(n+2\right)!+\left(n+1\right)!}{\left(n+2\right)!-\left(n+1\right)!}$ is

(a) 0
(b) −1
(c) 1
(d) none of these

Q.

The value of $\underset{x\to 0}{lim}\frac{(1+x{)}^{1/x}-e+\frac{1}{2}ex}{x^2}$ is

1. 

2. 

3. 

4. 

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

Q. The value of $\underset{x\to \infty }{lim}{\left(\frac{p^{1/x}+q^{1/x}+r^{1/x}+s^{1/x}}{4}\right)}^{3x},$ where $p,q,r,s>0$ is equal to

1. $pqrs$
2. $(pqrs{)}^3$
3. $(pqrs{)}^{3/2}$
4. $(pqrs{)}^{3/4}$

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.

$li{m}_{x\to 0}$ $\frac{1-cos2x}{x}$
[MMR 1983]

1. 0

2. 1

3. 2

4. 4

Q. If $\alpha$ is a repeated root of $a{x}^2+bx+c=0$ then $\underset{x\to \alpha }{lim}\frac{\sin (a{x}^2+bx+c)}{(x-\alpha {)}^2}$ is

1. $0$
2. $a$
3. $b$
4. $c$

Q. sin ax htlim sínar, ab * O14.

Q.

$\underset{x\to \infty }{lim}\frac{sinax}{bx}=$

1. 

2. 0

3. 

4. 1

Q.

If 0 < a < b, then $\underset{n\to \infty }{lim}(b^n+a^n{)}^{1/n}$ is equal to

1. e

2. a

3. b

4. 1