Mathematics : Limits and Derivatives

$\underset{\mathrm{x}\to 0}{\lim }\frac{6^{\mathrm{x}} - 3^{\mathrm{x}} - 2^{\mathrm{x}} + 1}{{\mathrm{x}}^2} = ?$

• $\left({\log }_{\mathrm{e}}\left(2\right)\right){\log }_{\mathrm{e}}\left(3\right)$

• ${\log }_{\mathrm{e}}\left(5\right)$

• ${\log }_{\mathrm{e}}\left(6\right)$

• 0

$$\begin{gathered} \mathrm{Define} \mathrm{f}\left(\mathrm{x}\right) =\left\{ {\mathrm{x}}^2 + \mathrm{bx} + \mathrm{c}, \mathrm{x} < 1 \\ \mathrm{x}, \mathrm{x} \ge 1\right\} \mathrm{If} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{differentiable} \mathrm{at} \mathrm{x} = 1, \mathrm{then} \\ \left(\mathrm{b} - \mathrm{c}\right) = ? \end{gathered}$$

•  - 2

• 0

• 1

• 2

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

• $\frac{1}{\mathrm{k}}$

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

• $\frac{1}{\mathrm{k} + 1}$

• $\frac{2}{\mathrm{k}}$

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

• $\frac{ - 1}{4}$

• $\frac{1}{2}$

• 1

• 2

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

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

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

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

• $\frac{\mathrm{\pi }}{3}$

If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0)is equal to

•  - 24

•  - 12

• 6

• 12

If a function f(x) defined by

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{{\mathrm{ae}}^{\mathrm{x}} + {\mathrm{be}}^{ - \mathrm{x}}, - 1 \le \mathrm{x} < 1 \\ {\mathrm{cx}}^2, 1 \le \mathrm{x} \le 3 \\ {\mathrm{ax}}^2 2\mathrm{cx}, 3 < \mathrm{x} \le 4\right\} \\ \mathrm{be} \mathrm{continuous} \mathrm{for} \mathrm{some} \mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathrm{R} \mathrm{and} \mathrm{f},\left(0\right) + \mathrm{f}\text{'}\left(2\right) = \mathrm{e}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{a} \mathrm{is} : \end{gathered}$$

• $\frac{1}{{\mathrm{e}}^2 - 3\mathrm{e} + 13}$

• $\frac{\mathrm{e}}{{\mathrm{e}}^2 - 3\mathrm{e} + 13}$

• $\frac{\mathrm{e}}{{\mathrm{e}}^2 - 3\mathrm{e} - 13}$

• $\frac{\mathrm{e}}{{\mathrm{e}}^2 + 3\mathrm{e} + 13}$

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

$\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

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