Right Hand Limit

Q.

Give the Maclaurin series for $\sin x$.

Q. $\text{Let}f\left(x\right)=\frac{1-x(1+|1-x|)}{|1-x|}\cos \left(\frac{1}{1-x}\right)$ for $x\ne 1.$ Then

1. $\underset{x\to 1^{-}}{lim}f\left(x\right)=0$
2. $\underset{x\to 1^{-}}{lim}f\left(x\right)$ does not exist
3. $\underset{x\to 1^{+}}{lim}f\left(x\right)=0$
4. $\underset{x\to 1^{+}}{lim}f\left(x\right)$ does not exist

Q.

How do you find the exact value of ${\sin }^{-1}0$?

Q.

If $a,b,c$ are in A.P., then $3^{\mathrm{a}},3^{\mathrm{b}},3^{\mathrm{c}}$ shall be in

Q.

If $y=e^x·e^{x^2}·e^{x^3}\dots e^{x^n}\dots .$, for $0<x<1$, then $\frac{dy}{dx}$at $x=\frac{1}{2}$ is:

1. $e$

2. $4e$

3. $2e$

4. $3e$

Q. The value of $\underset{x\to 1}{lim}\frac{100}{1-x^{100}}-\frac{50}{1-x^{50}}$ is

Q. Evaluate the limit:
$\underset{x\to 2}{lim}\frac{x-2}{{\log }_a(x-1)}$

Q.

Find ${lim}_{x\to 3^{+}}\frac{x}{\left[x\right]}$. Is it equal to ${lim}_{x\to 3^{-}}\frac{x}{\left[x\right]}$.

Q.

Find the coefficient of $a^3{b}^4{c}^5$ in ${(ab+bc+ca)}^6$.

1. $60$

2. $45$

3. $40$

4. $90$

Q.

Evaluate the following one sided limits:

$\left(i\right){lim}_{x\to 2^{+}}\frac{x-3}{x^2-4}$

$\left(ii\right){lim}_{x\to 2^{-}}\frac{x-3}{x^2-4}$

$\left(iii\right){lim}_{x\to 0^{+}}\frac{1}{3x}$

$\left(iv\right){lim}_{x\to 8^{+}}\frac{2x}{x+8}$

$\left(v\right){lim}_{x\to 0^{+}}\frac{2}{x^{\frac{1}{5}}}$

$\left(vi\right){lim}_{x\to \frac{{\pi }^{-}}{2}}tanx$

$\left(vii\right){lim}_{x\to \frac{\pi }{2^{+}}}secx$

$\left(viii\right){lim}_{x\to 0^{-}}\frac{x^2-3x+2}{x^3-2x^2}$

$\left(ix\right){lim}_{x\to -2^{+}}\frac{x^2-1}{2x+4}$

$\left(x\right){lim}_{x\to 0^{+}}(2-cotx)$

$\left(xi\right){lim}_{x\to 0^{-}}1+cosecx$

Q.

The value of $\cos 66+\sin 84$ is

Q.

Which of the following functions is decreasing on $\left(0,\frac{\pi }{2}\right)$?

1. $\sin 2x$

2. $\cos 3x$

3. $\tan x$

4. $\cos 2x$

Q.

What is the limit as $\mathrm{x}$ approaches infinity of ${\mathrm{e}}^{\mathrm{x}}$ ?

Q. If $\underset{x\to 3}{lim}{\left(\frac{\sqrt{2x+3}-x}{\sqrt{x+1}-x+1}\right)}^{\frac{x-1-\sqrt{x^2-5}}{x^2-5x+6}}$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a,b,c\in N,$ then the least value of $a^2+b^2+c^2$ is

Q.

The coefficient of $x^7$ in the expression ${(1+x)}^{10}+x{(1+x)}^9+x^2{(1+x)}^8+········+x^{10}$is

1. $420$

2. $330$

3. $210$

4. $120$

Q.

Solve the following:

$25÷(625÷5)$

Q. $\underset{n\to \infty }{lim}[\frac{1}{1+n^2}+\frac{2}{2^2+n^2}+...+\frac{n}{n^2+n^2}]$ is

1. $log2$
2. $\frac{\log 2}{2}$
3. $2log2$
4. $\frac{\log 2}{3}$

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

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

Q. The derivative of ${\log }_{10}x$ with respect to $x^2$ is

1. $\frac{1}{2x^2}{\log }_{e}10$
2. $\frac{2}{x^2}{\log }_{10}e$
3. $\frac{1}{2x^2}{\log }_{10}e$
4. none of these

Q. Find an anti derivative (or integral) of the given function by the method of inspection. $e^{2x}$

Q.

${lim}_{x\to 3}\frac{x-3}{|x-3|},$ is equal to

1. -1

2. does not exist

3. 1

4. 0

Q.

The coefficient of $x^{100}$ in the expression of ${\displaystyle \sum _{j=0}^{200}}{(1+x)}^j$ is

1. ${}^{200}C_{100}$

2. ${}^{201}C_{102}$

3. ${}^{200}C_{101}$

4. ${}^{201}C_{100}$

Q. If $f\left(x\right)=\frac{2}{x-3}$, $g\left(x\right)=\frac{x-3}{x+4}$ and $h\left(x\right)=-\frac{2(2x+1)}{x^2+x-12}$, then $\underset{x\to 3}{lim}\left[f\right(x)+g(x)+h(x\left)\right]$ is

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

Q. If $x,4$ and $\left[y\right]$ are the three terms of a G.P., where $x$ is an integer in the domain of $f\left(x\right)={\sin }^{-1}\frac{x}{2}$ and $\left[y\right]$ denotes the greatest integer less than or equal to $y$, then identify the correct statement(s).

1. Number of such G.P. is $4$.
2. Infinite number of G.P. is possible.
3. The set of values of $y$ is $[-16,-15)\cup [-8,-7)\cup [8,9)\cup [16,17)$
4. The set of values of $y$ is $[8,9)\cup [16,17)$

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

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}\underset{x\to \infty }{lim}(\frac{x^2+1}{x+1}-ax-b)=0,\text{then}& \text{(P)}& a=\frac{3}{2},b\in R\\ \text{(B)}& \text{If}\underset{x\to 0}{lim}(1+ax+b{x}^2{)}^{2/x}=e^3,\text{then}& \text{(Q)}& a=1,b=-\frac{1}{2}\\ \text{(C)}& \text{If}\underset{x\to 0}{lim}\left(\frac{a{e}^x-b}{x}\right)=2,\text{then}& \text{(R)}& a=1,b=-1\\ \text{(D)}& \text{If}\underset{x\to \infty }{lim}\{\sqrt{(x^2-x+1)}-ax-b\}=0,\text{then}& \text{(S)}& a=2,b=2\end{array}$

Which of the following is the only CORRECT combination?

1. $\left(B\right)\to \left(P\right),\left(C\right)\to \left(S\right)$
2. $\left(B\right)\to \left(Q\right),\left(C\right)\to \left(R\right)$
3. $\left(B\right)\to \left(R\right),\left(C\right)\to \left(Q\right)$
4. $\left(B\right)\to \left(S\right),\left(C\right)\to \left(S\right)$

Q. The value of ${}^6{C}_4$ is

1. $9$
2. $15$
3. $240$
4. $6$

Q.

${lim}_{x\to 0^{+}}{\{1+{\tan }^2\sqrt{x}\}}^{\frac{1}{2}x}$

Q.

If $f\left(x\right)=\left\{\begin{array}{ll}\left|x\right|,& 0<\left|x\right|\le 2\\ 1,& x=0\end{array}\right.$, then at $x=0$, $f$ has

1. a local maximum.

2. a local minimum.

3. no local extremum.

4. no local maximum.

Q.

${\int }_0^2{e}^{x}dx$

Q. Integrate the function $e^x(\frac{1}{x}-\frac{1}{x^2})$