Limits and Derivatives

$\underset{\mathrm{n}\to \infty }{\lim }\left[\frac{1^{\mathrm{k}} + 2^{\mathrm{k}} + 3^{\mathrm{k}} + ... + {\mathrm{n}}^{\mathrm{k}}}{{\mathrm{n}}^{\mathrm{k} + 1}}\right] = ?$

• $\frac{1}{\mathrm{k}}$

• $\frac{2}{\mathrm{k} \hspace{0.17em}+ 1}$

• $\frac{1}{\mathrm{k} + 1}$

• $\frac{2}{\mathrm{k}}$

$\underset{\mathrm{x}\to 0}{\lim } \frac{\left(1 - \cos \left(2\mathrm{x}\right)\right)\left(3 + \cos \left(\mathrm{x}\right)\right)}{\mathrm{xtan}\left(4\mathrm{x}\right)} = ?$

• $\frac{ - 1}{4}$

• $\frac{1}{2}$

• 1

• 2

$\mathrm{An} \mathrm{angle} \mathrm{between} \mathrm{the} \mathrm{curves} {\mathrm{x}}^2=3\mathrm{y} \mathrm{and} {\mathrm{x}}^2 + {\mathrm{y}}^2 = 4 \mathrm{is}$

• ${\tan }^{-1}\left(\frac{5}{\sqrt{3}}\right)$

• ${\tan }^{-1}\left(\sqrt{\frac{5}{3}}\right)$

• ${\tan }^{-1}\left(\frac{2}{\sqrt{3}}\right)$

• $\frac{\mathrm{\pi }}{3}$

$\mathrm{If} \underset{\mathrm{x}\to 1}{\lim }\frac{\mathrm{x} +{\mathrm{x}}^2 + {\mathrm{x}}^3 +\hspace{0.17em}... + {\mathrm{x}}^{\mathrm{n}} - \mathrm{n}}{\mathrm{x} - 1} = 820, \left(\mathrm{n} \in \mathrm{N}\right) \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{n} =?$

$\underset{\mathrm{x}\to 0}{\lim }{\left(\tan \left(\frac{\mathrm{\pi }}{4} + \mathrm{x}\right)\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.} = ?$

• 2

• 1

• e

• e²

$\underset{\mathrm{x}\to 0}{\lim }\frac{\left(1 - \cos \left({\displaystyle \frac{{\mathrm{x}}^2}{2}}\right)\right)\left(1 - \cos \left({\displaystyle \frac{{\mathrm{x}}^2}{4}}\right)\right)}{{\mathrm{x}}^8} = 2^{ - \mathrm{k}}, \mathrm{find} \mathrm{k}$

$\mathrm{The} \mathrm{value} \mathrm{of} 0 . {16}^{{\log }_{2 . 5}\left(\frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + ... \infty \right)} \mathrm{is}$

$\underset{\mathrm{x}\to \mathrm{a}}{\lim }\frac{{\left(\mathrm{a} + 2\mathrm{x}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}} - {\left(3\mathrm{x}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}{{\left(3\mathrm{a} + \mathrm{x}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}} - {\left(4\mathrm{x}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}} \left(\mathrm{a} \ne 0\right) = ?$

• $\left(\frac{2}{9}\right){\left(\frac{2}{3}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

• ${\left(\frac{2}{3}\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

• ${\left(\frac{2}{9}\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

• $\left(\frac{2}{3}\right){\left(\frac{2}{9}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$$\begin{gathered} \mathrm{Let} \mathrm{f} : \left(0, \infty \right) \to \left(0, \infty \right) \mathrm{be} \mathrm{a} \mathrm{differentiable} \mathrm{function} \mathrm{such} \mathrm{that} \mathrm{f}\left(1\right) = \mathrm{e} \mathrm{and} \\ \underset{\mathrm{t}\to \mathrm{x}}{\lim } \frac{{\mathrm{t}}^2{\mathrm{f}}^2\left(\mathrm{x}\right) - {\mathrm{x}}^2{\mathrm{f}}^2\left(\mathrm{t}\right)}{\mathrm{t} - \mathrm{x}}. \mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = 1, \mathrm{then} \mathrm{x} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• $\frac{1}{\mathrm{e}}$

• 2e

• $\frac{1}{2\mathrm{e}}$

• e

$\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{x}\left({\mathrm{e}}^{\left({}^{\sqrt{1 + {\mathrm{x}}^2 + {\mathrm{x}}^4} - 1}\right)/\mathrm{x}} - 1\right)}{\sqrt{1 + {\mathrm{x}}^2 + {\mathrm{x}}^4} - 1}$

• does not exist

• is equal to 1

• is equal to $\sqrt{\mathrm{e}}$

• is equal to 0