The value of $\underset{\mathrm{n}\to \infty }{\lim }\left[\frac{\mathrm{n}}{{\mathrm{n}}^2 + 1^2} + \frac{{\displaystyle \mathrm{n}}}{{\displaystyle {\mathrm{n}}^2 + 2^2}} + ... + \frac{{\displaystyle \mathrm{n}}}{{\displaystyle {\mathrm{n}}^2 + {\mathrm{n}}^2}}\right]$ is

• $\frac{\mathrm{\pi }}{4}$

• $\log \left(2\right)$

• 0

• 1

The value of $\underset{\mathrm{n}\to \infty }{\lim }\left(\frac{1}{\mathrm{n} + 1} + \frac{1}{\mathrm{n} + 2} + ... + \frac{1}{6\mathrm{n}}\right)$ is

• $\log \left(2\right)$

• $\log \left(6\right)$

• 1

• $\log \left(3\right)$

$\underset{\mathrm{x}\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{{\mathrm{a}}^{\cot \left(\mathrm{x}\right)} - {\mathrm{a}}^{\cos \left(\mathrm{x}\right)}}{\cot \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right)}$

• ${\log }_{\mathrm{e}}\left(\frac{\mathrm{\pi }}{2}\right)$

• ${\log }_{\mathrm{e}}\left(2\right)$

• ${\log }_{\mathrm{e}}\left(\mathrm{a}\right)$

• a

The value of $\underset{\mathrm{x}\to 2}{\lim }\frac{5}{\sqrt{2} - \sqrt{\mathrm{x}}}$ is

• 10$\sqrt{2}$

• + $\infty$

• - $\infty$

• does not exist

The value of $\underset{\mathrm{x}\to 2}{\lim }\frac{{\mathrm{e}}^{3\mathrm{x} - 6} - 1}{\sin \left(2 - \mathrm{x}\right)}$

• $\frac{3}{2}$

• 3

• - 3

• - 1

$\frac{d^{\mathrm{n}}}{d{\mathrm{x}}^{\mathrm{n}}}\left(\log \left(\mathrm{x}\right)\right)$ is equal to

• $\frac{\left(\mathrm{n} - 1\right)!}{{\mathrm{X}}^{\mathrm{n}}}$

• $\frac{\mathrm{n} !}{{\mathrm{X}}^{\mathrm{n}}}$

• $\frac{\left(\mathrm{n} - 2\right)!}{{\mathrm{X}}^{\mathrm{n}}}$

• ${\left(- 1\right)}^{\mathrm{n} - 1}\frac{\left(\mathrm{n} - 1\right)!}{{\mathrm{X}}^{\mathrm{n}}}$

47.

The value of $\underset{\mathrm{x}\to \infty }{\lim }\sqrt{{\mathrm{a}}^2{\mathrm{x}}^2 + \mathrm{ax} + 1} - \sqrt{{\mathrm{a}}^2{\mathrm{x}}^2 + 1}$ is

• $\frac{1}{2}$

• 1

• 2

• None of these

$\underset{\mathrm{x}\to 0}{\lim }\frac{\left(1 - \cos \left(2\mathrm{x}\right)\right)\sin \left(5\mathrm{x}\right)}{{\mathrm{x}}^2\sin \left(3\mathrm{x}\right)}$ equals

• $\frac{10}{3}$

• $\frac{3}{10}$

• $\frac{6}{5}$

• $\frac{5}{6}$

$\underset{\mathrm{x}\to \infty }{\lim }{\left(1 - \frac{4}{\mathrm{x} - 1}\right)}^{3\mathrm{x} - 1}$ is equal to

• e¹²

• e^{- 12}

• e⁴

• e³

$\underset{\mathrm{x}\to 0}{\lim }\frac{{\mathrm{a}}^{\mathrm{x}} - {\mathrm{b}}^{\mathrm{x}}}{{\mathrm{e}}^{\mathrm{x}} - 1}$ is equal to 

• ${\log }_{\mathrm{e}}\left(\frac{\mathrm{a}}{\mathrm{b}}\right)$

• ${\log }_{\mathrm{e}}\left(\frac{\mathrm{b}}{\mathrm{a}}\right)$

• ${\log }_{\mathrm{e}}\left(\mathrm{ab}\right)$

• ${\log }_{\mathrm{e}}\left(\mathrm{a} + \mathrm{b}\right)$