Standard Limits to Remove Indeterminate Form

Q. In a triangle $ABC$ with usual notation, which of the following is (are) CORRECT?

1. If the angles $A,B,C$ are in A.P., then $b^2=c^2+a^2-ca$
2. $\frac{\sin (B-C)}{\sin (B+C)}=\frac{b^2-c^2}{a^2}$
3. $\sum \frac{b^2-c^2}{\cos B+\cos C}=0$
4. $\frac{1+\cos (A-B)\cos C}{1+\cos (A-C)\cos B}=\frac{a^2+b^2}{a^2+c^2}$

Q. If $\underset{x\to 0}{lim}\frac{a\tan 3x+(1-\cos 2x)}{x+\sin x+\tan x}=1$, then the value of $a$ is

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

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

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.

$li{m}_{x\to 0}\frac{(e^{1/x}-1)}{(e^{1/x}+1)}$

1. 0

2. 1

3. -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.

$\underset{x\to 0}{lim}\frac{1-cosxcos2xcos3x}{si{n}^{2}2x}$ is equal to

1. $\frac{7}{2}$

2. $\frac{7}{3}$

3. $\frac{7}{4}$

4. $\frac{7}{5}$

Q. $\underset{x\to 0}{lim}\frac{\sin \left(x^2\right)}{\ln \left(\cos \right(2x^2-x\left)\right)}$ is equal to

1. $2$
2. $-2$
3. $1$
4. $-1$

Q.

$li{m}_{x\to 0}$ $\frac{\left(x^{3}cotx\right)}{1-cosx}$ =


[AI CBSE ; DSSE 1988]

1. 0

2. 1

3. 2

4. -2

Q.

$\underset{x\to 0}{lim}\frac{1-cosxcos2xcos3x}{si{n}^{2}2x}$ is equal to

1. $\frac{7}{2}$

2. $\frac{7}{3}$

3. $\frac{7}{4}$

4. $\frac{7}{5}$

Q.

$\underset{x\to 0}{lim}$ $\frac{x(e^x-1)}{1-cosx}$ =

1. 0

2. ∞

3. -2

4. 2

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. ${lim}_{x\to 0}\frac{sin\left(\pi co{s}^{2}x\right)}{x^2}$ equals

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

Q. If $n$ is the number of real solutions of the equation $min(e^{-|x|},1-e^{-|x|})=\frac{1}{4}$ and $L=\underset{x\to 0^{-}}{lim}(\frac{e^{2x}-1}{x}+\frac{e^{3x}-1}{x}+\frac{e^{4x}-1}{x}+\cdots \text{upto}n\text{terms}),$ then the value of $L$ is

1. $5$
2. $7$
3. $10$
4. $14$

Q.

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

1. 1

2. 2

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

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

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

Q. Value of $\underset{x\to \infty }{\text{lim}}{\left(\frac{x^2+2x-1}{2x^2-3x-2}\right)}^{\frac{2x+1}{2x-1}}$ is

1. It does not exist
2. $0$
3. $-2$
4. $\frac{1}{2}$

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

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

Q. If $n$ is the number of real solutions of the equation $min(e^{-|x|},1-e^{-|x|})=\frac{1}{4}$ and $L=\underset{x\to 0^{-}}{lim}(\frac{e^{2x}-1}{x}+\frac{e^{3x}-1}{x}+\frac{e^{4x}-1}{x}+\cdots \text{upto}n\text{terms}),$ then the value of $L$ is

1. $5$
2. $7$
3. $10$
4. $14$

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.

$li{m}_{x\to 0}$ $\frac{\left(x^{3}cotx\right)}{1-cosx}$ =


[AI CBSE ; DSSE 1988]

1. 0

2. 1

3. 2

4. -2

Q. $\text{If}\underset{x\to -1}{lim}{\left(\frac{\tan (x+1)+a(e.e^x+x)}{x-a^2{\sin }^{-1}(x+1)+1}\right)}^{\frac{x+1}{\sqrt{x+2}-1}}=1,$
$\text{then possible values of}{}^{\prime }{a}^{\prime }\text{is/are}$

1. $-2$
2. $0$
3. $\sqrt{3}+1$
4. $\sqrt{3}-1$

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

Q.

$\text{The value of}{lim}_{x\to \infty }{\left(\frac{x^2-1}{x^2+1}\right)}^{x^2}is$

1. 1

2. $e^{-1}$

3. $e^{-2}$

4. $e^{-3}$

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

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

Q. $\underset{x\to 0}{lim}\frac{(1-\cos 2x)(3+\cos 3x)}{x\tan 4x}$ is equal to :

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

Q.

The value of ${lim}_{x\to \infty }x[ta{n}^{-1}\left(\frac{x+1}{x+2}\right)-ta{n}^{-1}\left(\frac{x}{x+2}\right)]is$

1. 1

2. -1

3. 1/2

4. -1/2

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. $\underset{x\to 0}{lim}\frac{\sin \left(\pi {\cos }^{2}x\right)}{x^2}$ is equal to:

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

Q.

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

1. 1

2. -1

3. 0

4. 2