$\mathrm{If} \mathrm{a} > 0, \underset{\mathrm{x}\to \mathrm{a}}{\lim } \frac{{\mathrm{a}}^{\mathrm{x}} - {\mathrm{x}}^{\mathrm{a}}}{{\mathrm{x}}^{\mathrm{x}} - {\mathrm{a}}^{\mathrm{a}}} = - 1, \mathrm{then} \mathrm{a} \mathrm{is} \mathrm{equal} \mathrm{to} :$

• 0

• 1

• e

• 2e

The value of $\underset{\mathrm{n}\to \infty }{\lim }\frac{1}{{\mathrm{n}}^3}\sum _{\mathrm{k}=1}^{\mathrm{n}}\left({\mathrm{k}}^2\mathrm{x}\right)$ is

• x

• $\frac{\mathrm{x}}{2}$

• $\frac{\mathrm{x}}{3}$

• $\frac{\mathrm{x}}{4}$

If f: $\mathrm{R} \to \mathrm{R}$ is an even function having derivatives of all orders, then an odd function among the following is

• f''

• f'''

• f' + f''

• f'' + f'''

If x > 0, x^y = e^{x - y}, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{1}{{\left(1 + \log \left(\mathrm{x}\right)\right)}^2}$

• $\frac{\log \left(\mathrm{x}\right)}{{\left(1 + \log \left(\mathrm{x}\right)\right)}^2}$

• ${\left(\frac{\log \left(\mathrm{x}\right)}{{\left(1 + \log \left(\mathrm{x}\right)\right)}^2}\right)}^2$

• $\frac{{\left(\log \left(\mathrm{x}\right)\right)}^2}{1 + \log \left(\mathrm{x}\right)}$

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

• 1

• 0

• does not exist

• $\infty$

$\mathrm{If} 0 < \mathrm{p} <\mathrm{q}, \mathrm{then} \underset{\mathrm{n}\to \infty }{\lim }{\left({\mathrm{q}}^{\mathrm{n}} + {\mathrm{p}}^{\mathrm{n}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{n}$}\right.} = ?$

• e

• p

• q

• 0

197.

$\underset{\mathrm{x}\to \infty }{\lim }\left[\sqrt{{\mathrm{x}}^2 + 2\mathrm{x} - 1} - \mathrm{x}\right] = ?$

• $\infty$

• $\frac{1}{2}$

• 4

• 1

$$\begin{gathered} \mathrm{If} {\mathrm{l}}_1 = \underset{\mathrm{x}\to 2^{ + }}{\lim }\left(\mathrm{x} + \left[\mathrm{x}\right]\right), {\mathrm{l}}_2 = \underset{\mathrm{x}\to 2^{ - }}{\lim }\left(2\mathrm{x} - \left[\mathrm{x}\right]\right) \\ \mathrm{and} {\mathrm{l}}_3 = \underset{\mathrm{x}\to \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\lim }\frac{\cos \left(\mathrm{x}\right)}{\left(\mathrm{x} - {\displaystyle \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}, \mathrm{then} : \end{gathered}$$

• ${\mathrm{l}}_1 < {\mathrm{l}}_2 < {\mathrm{l}}_3$

• ${\mathrm{l}}_2 < {\mathrm{l}}_3 < {\mathrm{l}}_1$

• ${\mathrm{l}}_3 < {\mathrm{l}}_2 < {\mathrm{l}}_1$

• ${\mathrm{l}}_1 < {\mathrm{l}}_3 < {\mathrm{l}}_2$

If $\underset{\mathrm{x}\to 0}{\lim }\left(\frac{\cos \left(4\mathrm{x}\right) + \mathrm{acos}\left(2\mathrm{x}\right) + \mathrm{b}}{{\mathrm{x}}^4}\right)$ is finite, then the values of a, b are respectively :

• 5, - 4

• - 5, - 4

• - 4, 3

• 4, 5

$\underset{\mathrm{x}\to 0}{\lim } \frac{{\mathrm{e}}^{\mathrm{x}} - {\mathrm{e}}^{\sin \left(\mathrm{x}\right)}}{2\left(\mathrm{x} - \sin \left(\mathrm{x}\right)\right)}$

• $- \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• 1

• $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$