Right Hand Derivative

Q. If $\underset{x\to \infty }{lim}(\frac{x^2+x+1}{x+1}-ax-b)=4$, then

1. $a=1,b=4$
2. $a=1,b=-4$
3. $a=2,b=-3$
4. $a=2,b=3$

Q. Let $f:R\to R$ be defined as $f\left(x\right)=\{\begin{array}{ccc}\frac{x^3}{(1-\cos 2x{)}^2}\cdot {\log }_e\left(\frac{1+2x{e}^{-2x}}{(1-x{e}^{-x}{)}^2}\right)& ,& x\ne 0\\ \alpha & ,& x=0\end{array}$
If $f$ is continuous at $x=0$, then $\alpha$ is equal to:

1. $0$
2. $1$
3. $2$
4. $3$

Q. If the function $f\left(x\right)=\{\begin{array}{ccc}{k}_1{(x-\pi )}^2-1,& x\le \pi & \\ k_2\cos x,& x>\pi & \end{array}$ is twice differentiable, then the ordered pair $(k_1,k_2)$ is equal to:

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

Q. Let $f\left(x\right)=x^6+2x^4+x^3+2x+3,x\in R.$ Then the natural number $n$ for which $\underset{x\to 1}{lim}\frac{x^{n}f\left(1\right)-f\left(x\right)}{x-1}=44$ is

Q. If $U_n=(1+\frac{1}{n^2}){(1+\frac{2^2}{n^2})}^2\dots {(1+\frac{n^2}{n^2})}^n$, then $\underset{n\to \infty }{lim}(U_n{)}^{\frac{-4}{n^2}}$ is equal to

1. $\frac{4}{e^2}$
2. $\frac{4}{e}$
3. $\frac{16}{e^2}$
4. $\frac{e^2}{16}$

Q. The value of $\underset{n\to \infty }{lim}[\frac{1}{\sqrt{(2n-1^2)}}+\frac{1}{\sqrt{(4n-2^2)}}+\frac{1}{\sqrt{(6n-3^2)}}+...+\frac{1}{n}]$ is equal to

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

Q. Let $K$ be the set of all real values of $x$ where the function $f\left(x\right)=\sin |x|-|x|+2(x-\pi )\cos |x|$ is not differentiable. Then the set $K$ is equal to :

1. $\left\{0\right\}$
2. $\left\{\pi \right\}$
3. $\varphi$ (an empty set)
4. $\{0,\pi \}$

Q. If $\underset{x\to \infty }{lim}{(e^{2x}+e^x+x)}^{\frac{1}{x}}=e^a$, then the value of $a$ is

Q. If $f:R\to R$ is given by $f\left(x\right)=x+1,$ then the value of $\underset{n\to \infty }{lim}\frac{1}{n}\left[f\right(0)+f\left(\frac{5}{n}\right)+f\left(\frac{10}{n}\right)+...+f\left(\frac{5(n-1)}{n}\right)],$ is:

1. $\frac{7}{2}$
2. $\frac{3}{2}$
3. $\frac{5}{2}$
4. $\frac{1}{2}$

Q. Let $[.]$ denote the greatest integer function.
Consider $f\left(x\right)=\{\begin{array}{cc}2-\left|x\right|,& -1\le x\le 1\\ |x-2|-x,& 1<x\le 3\end{array}$ and $g\left(x\right)=\{\begin{array}{cc}\sin x-1,& 0\le x<\frac{\pi }{2}\\ \left[x\right]-\cos (x-2),& \frac{\pi }{2}\le x\le \pi \end{array}.$

Which of the following statements is/are CORRECT?

1. $\underset{x\to 1^{+}}{lim}g\left(f\right(x\left)\right)=-1$
2. $\underset{x\to \pi /2^{-}}{lim}g\left(f\right(g\left(x\right)\left)\right)=0$
3. $\underset{x\to 2^{+}}{lim}\frac{f\left(g\right(x\left)\right)}{f\left(x\right)-2}=\frac{1}{2}$
4. $\underset{x\to 0^{+}}{lim}\frac{g\left(f\right(x\left)\right)}{\left(f\right(x)-2{)}^2}=\frac{1}{2}$

Q. The value of $\underset{n\to \infty }{lim}(\frac{1}{\sqrt{n^2}}+\frac{1}{\sqrt{n^2-1^2}}+\frac{1}{\sqrt{n^2-2^2}}+\cdots +\frac{1}{\sqrt{n^2-(n-1{)}^2}})$ is

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

Q. $\underset{x\to \infty }{lim}{\left(\frac{a^{1/x}+b^{1/x}+c^{1/x}}{3}\right)}^x=$ (where $a,b,c$ are positive real numbers)

1. $0$
2. $(abc{)}^{1/3}$
3. $(abc{)}^{-1/3}$
4. $1$

Q. $\underset{x\to 0}{lim}\frac{x}{a}\left[\frac{b}{x}\right](a\ne 0),$ where $[\cdot ]$ denotes the greatest integer function, is equal to

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

Q. The value of $\underset{x\to a}{lim}(\frac{|x{|}^3}{a}-{\left[\frac{x}{a}\right]}^3),a>0$
(where $[.]$ denotes the greatest integer function) is

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

Q. Let $f\left(x\right)=\frac{{(27-2x)}^{1/3}-3}{9-3{(243+5x)}^{1/5}},x\ne 0.$ If $f\left(x\right)$ is continuous at $x=0$, then the value of $f\left(0\right)$ is

1. $2/3$
2. $2$
3. $4$
4. $6$

Q. Let $\lambda =\underset{0}{\overset{1}{\int }}\frac{dx}{1+x^3}$ and $p=\underset{n\to \infty }{lim}{\left(\frac{\prod _{r=1}^n(n^3+r^3)}{n^{3n}}\right)}^{1/n}.$ Then the value of $\ln p$ is equal to

1. $\ln 2-1+3\lambda$
2. $\ln 2-3+3\lambda$
3. $2\ln 2+3\lambda$
4. $\ln 4-3+3\lambda$

Q. Let $\underset{h\to 0}{lim}\frac{h^2}{f(x+2h)-2f(x+h)+f\left(x\right)}=\frac{x^{1-x}}{1+x(1+\ln x{)}^2}.$ If $\underset{x\to 0^{+}}{lim}f\left(x\right)=1$ and $f\left(1\right)=2,$ then the value of $\frac{f\left(3\right)}{f\left(2\right)}$ is

Q. Let $f\left(x\right)=\left\{\begin{array}{cc}\underset{0}{\overset{x}{\int }}(5+|1-t|)dt,& x>2\\ 5x+1,& x\le 2\end{array}\right\}$
Which of the following is true for $f\left(x\right)$?

1. The right hand derivative of $f\left(x\right)$ at $x=2$ doesn’t exist
2. $f\left(x\right)$ is continuous at $x=2$
3. $f\left(x\right)$ is continuous but not differentiable at $x=2$
4. $f\left(x\right)$ is everywhere differentiable

Q. Let $f\left(x\right)=\frac{1}{1+e^{1/x}}$ for $x\ne 0$ and $f\left(0\right)=0.$ Then

1. $f\left(0^{+}\right)$ does not exist
2. $f\left(0^{-}\right)$ is equal to zero
3. $f^{\prime }\left(0^{-}\right)$ is equal to $1$
4. $f^{\prime }\left(0^{+}\right)$ is equal to zero

Q. If $f\left(x\right)=\{\begin{array}{cc}\frac{\sin (a+2)x+\sin x}{x},& x<0\\ b,& x=0\\ \frac{(x+3x^2{)}^{1/3}-x^{1/3}}{x^{4/3}},& x>0\end{array}$ is continuous at $x=0,$ then the value of $a+2b$ is

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

Q. $\underset{x\to 0^{-}}{lim}\left(\frac{5x+2|x|}{7x+3|x|}\right)$ is equal to

1. $\frac{3}{4}$
2. $\frac{7}{10}$
3. $0$
4. $\infty$

Q. Let $f\left(x\right)=\{\begin{array}{cc}x+e^{2x}-1,& x<0\\ x^2+2\lambda x,& x\ge 0\end{array}.$ If $f\left(x\right)$ is differentiable at $x=0,$ then the value of $2\lambda$ is

Q. The value of $\underset{n\to \infty }{lim}\frac{\left[\right(n+1\left)\right(n+2)...(n+n){]}^{\frac{1}{n}}}{n}$ is:

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

Q. If $\underset{x\to 0}{lim}\frac{a{e}^x-b\cos x+c{e}^{-x}}{x\sin x}=2,$, then $a+b+c$ is equal to

Q. Let $\left[t\right]$ denote the greatest integer $\le t$. If for some $\lambda \in R-\{0,1\}$, $\underset{x\to 0}{lim}\left|\frac{1-x+\left|x\right|}{\lambda -x+\left[x\right]}\right|=L$, then $L$ is equal to

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

Q. The value of $\underset{n\to \infty }{lim}[\frac{1}{na}+\frac{1}{na+1}+\frac{1}{na+2}+\cdots +\frac{1}{nb}]$ where $a,b>0$ and $a\ne b$ is:

1. $\ln \left(\frac{b}{a}\right)$
2. $\ln \left(ab\right)$
3. $\ln \left(a^{2}b\right)$
4. $\ln (a+b)$

Q. Number of points where $f\left(x\right)=\left[x\right]{\sin }^2\left(\pi x\right)$ is not differentiable if $x\in (-7,10)$ is (where $[.]$ denotes the greatest integer function)

1. $15$
2. $0$
3. $6$
4. $17$

Q. Let $f\left(x\right)=\{\begin{array}{cc}\underset{n\to \infty }{lim}\frac{e^{x^2}-1+\left[\right(a+b)x-(a-b\left)\right]x^{2n}}{x^{2n+1}+\cos x-1},& x\in R-\left\{0\right\}\\ k,& x=0\end{array}$
If $f\left(x\right)$ is continuous for all $x\in R$, then the value $|k|$ is

Q. If $f\left(x\right)=9^{-2/x^2},x\ne 0$ is continuous at $x=0,$ then the value of $f\left(0\right)$ is

1. $0$
2. $1$
3. $9$
4. $-1$

Q. If the value of $\underset{n\to \infty }{lim}{\left(\frac{(2n+1)!}{n^{2n+1}}\right)}^{\frac{1}{n}}=\ln a+b,$ then the value of $(a-b)$ equals to (where $a,b\in Z$)