Indeterminate Forms

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. $\underset{x\to \infty }{lim}(\frac{1}{n^3+1}+\frac{4}{n^3+1}+\frac{9}{n^3+1}+\dots \dots +\frac{n^2}{n^3+1})$ is equal to

1. 
2. 
3. 
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.

If $\left\{p\right\}$ denotes the fractional part of the number $p$, then $\left\{\frac{3^{200}}{8}\right\}$ is equal to:

1. $\frac{5}{8}$

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

3. $\frac{7}{8}$

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

Q. Evaluate the limit:

$\underset{x\to 1}{lim}\frac{x^3+3x^2-6x+2}{x^3+3x^2-3x-1}$

Q.

The weight of a body on the surface of the earth is 12.6 N. When it is raised to a height half the radius of the earth, its weight will be

1. $2.8N$

2. $5.6N$

3. $12.5N$

4. $25.2N$

Q.

Let $f:\left(0,2\right)\to R$ be defined as $f\left(x\right)={\log }_2\left[1+\tan \left(\frac{\mathrm{\pi x}}{4}\right)\right]$. Then, $\underset{n\to \infty }{\lim }\left(\frac{2}{n}\right)\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)..............f\left(1\right)\right]$ is equal to ______

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 $\underset{x\to 0}{lim}{\left(\frac{f(-x)}{2x^3}\right)}^{\frac{1}{x}}=e^{-3},$ then

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

Q. If $L=\underset{x\to -\infty }{lim}\sqrt{25x^2-3x}+5x$, then the value of $\left[\frac{1}{L}\right]$ is
(where $[.]$ denotes greatest integer function)

Q.

The range of$f\left(x\right)=6^x+3^x+6^{-x}+3^{-x}+2$ is a subset of.....

1. $[-\infty ,-2)$

2. $(-2,3)$

3. $(6,\infty )$

4. $[6,\infty )$

Q.

$\underset{x\to 0}{lim}\frac{e^{\alpha x}-e^{\beta x}}{x}=$

1. 

2. 

3. 

4. 

Q.

${lim}_{x\to 0}(\cos x+a\sin bx{)}^{\frac{1}{x}}$

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

Q. Evaluate the limit:

$\underset{x\to 1}{lim}\frac{x^4-3x^3+2}{x^3-5x^2+3x+1}$

Q. $\underset{x\to \infty }{lim}[(x^2+\frac{1}{x})e^{1/x}-x-x^2]$ is equal to

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

Q.

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

1. 

2. 

3. 

4. 

Q. Evaluate the limit : $\underset{x\to 0}{lim}\frac{3x+1}{x+3}$

Q. Express the following in the form a +bi, where a and b are real numbers.

Q. Let $P_n={(1-\frac{1}{3})}^2{(1-\frac{1}{6})}^2\cdots {(1-\frac{2}{n(n+1)})}^2,$ where $n\ge 2(n\in N).$ If $\underset{n\to \infty }{lim}{P}_n=\frac{a}{b},$ where $a$ and $b$ are coprime, then the value of $a+b$ is

Q. $\underset{x\to 0}{lim}\frac{\sum _{r=1}^{100}\left[x^r\right]}{1+|x|}=$

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

Q. Differentiate from first principle:

$\left(xii\right)x^3+4x^2+3x+2$

Q.

$f\left(x\right)=\frac{a{x}^2+b}{x^2+1},{lim}_{x\to 0}f\left(x\right)=1and{lim}_{x\to \infty }f\left(x\right)=1,thenprovethatf(-2)=f\left(2\right)=1.$

Q. The value of the limit $\underset{x\to 0}{lim}(\frac{x}{\sin x}{)}^{\frac{6}{x^2}}$ is

1. $e$
2. $e^{-1}$
3. $e^{-\frac{1}{6}}$
4. $e^6$

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

1. 
2. 
3. 
4. 

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. Evaluate the limit: $\underset{x\to \frac{1}{4}}{lim}\frac{4x-1}{2\sqrt{x}-1}$

Q. ${lim}_{x\to \frac{\pi }{2}}\frac{[1-\tan \left(\frac{x}{2}\right)][1-\sin x]}{[1+\tan \left(\frac{x}{2}\right)][\pi -2x{]}^3}$ is -

Q. If $f\left(x\right)=\underset{n\to \infty }{lim}\frac{\ln (5-x)-x^n\sin nx}{1+x^{2n}},x>1,$ then $f\left(3\right)$ is equal to

1. $0$
2. $1$
3. $2$
4. not defined

Q. Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& M_r\text{is defined as}{M}_r=\left[\begin{array}{cc}(r-1)& \frac{1}{r}\\ 1& \frac{1}{(r-1{)}^2}\end{array}\right]\text{. Then}\underset{n\to \infty }{lim}{(det{M}_2+det{M}_3+\cdots +det{M}_n)}^{\ln n}\text{equals}& \text{(P)}& 16\\ \text{(B)}& \text{If}\left|\begin{array}{ccc}1& \cos \alpha & \cos \beta \\ \cos \alpha & 1& \cos \gamma \\ \cos \beta & \cos \gamma & 1\end{array}\right|=\left|\begin{array}{ccc}0& \cos \alpha & \cos \beta \\ \cos \alpha & 0& \cos \gamma \\ \cos \beta & \cos \gamma & 0\end{array}\right|,\text{then}{\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma \text{equals}& \text{(Q)}& 4\\ \text{(C)}& \text{If}A=\left[\begin{array}{cc}3& 1\\ -1& 1\end{array}\right]\text{and a matrix}C\text{is defined as}C=\left(BA{B}^{-1}\right)\left(B^{-1}{A}^{\text{T}}B\right),\text{where}|B|\ne 0,& \text{(R)}& 2\\ & \text{then}detC\text{is a square of natural number equal to}& & \\ \text{(D)}& \text{If}A=\left[\begin{array}{cc}1& 1\\ -1& 1\end{array}\right]\text{and}{A}^4=-\lambda I,\text{then}\lambda \text{equals}& \text{(S)}& 1\end{array}$

Which of the following is a CORRECT combination?

1. $\text{(C)}\to \text{(P)},\text{(D)}\to \text{(S)}$
2. $\text{(C)}\to \text{(Q)},\text{(D)}\to \text{(Q)}$
3. $\text{(C)}\to \text{(Q)},\text{(D)}\to \text{(R)}$
4. $\text{(C)}\to \text{(R)},\text{(D)}\to \text{(P)}$

Q. Evaluate the limit:
$\underset{x\to 0}{lim}\frac{a^x+a^{-x}-2}{x^2}$