Limits and Derivatives

$\underset{\mathrm{x}\to \infty }{\lim }{\left(\frac{\mathrm{x} + 5}{\mathrm{x} +\hspace{0.17em}2}\right)}^{\mathrm{x} + 3}$ equals

• e

• e²

• e³

• e⁵

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

• 0

• 1

• $\frac{1}{2}$

• $- \frac{1}{2}$

$$\begin{gathered} \mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \left|\mathrm{x}\right| + \left|\sin \left(\mathrm{x}\right)\right| \mathrm{for} \mathrm{x} \in \left(- \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right), \\ \mathrm{then} \mathrm{its} \mathrm{left} \mathrm{hand} \mathrm{derivative} \mathrm{at} \mathrm{x} = 0 \mathrm{is} \end{gathered}$$

• 0

• - 1

• - 2

• - 3

$\mathrm{If} \mathrm{u} = \mathrm{u}\left(\mathrm{x}, \mathrm{y}\right) = \sin \left(\mathrm{y} + \mathrm{ax}\right) - {\left(\mathrm{y} + \mathrm{ax}\right)}^2, \mathrm{then} \mathrm{it} \mathrm{implies}$

• ${\mathrm{u}}_{\mathrm{xx}} = {\mathrm{a}}^2 . {\mathrm{u}}_{\mathrm{yy}}$

• ${\mathrm{u}}_{\mathrm{yy}} = {\mathrm{a}}^2{\mathrm{u}}_{\mathrm{xx}}$

• ${\mathrm{u}}_{\mathrm{xx}} = - {\mathrm{a}}^2 . {\mathrm{u}}_{\mathrm{yy}}$

• ${\mathrm{u}}_{\mathrm{yy}} = - {\mathrm{a}}^2{\mathrm{u}}_{\mathrm{xx}}$

$\underset{\mathrm{x}\to \infty }{\lim }{\left(\frac{\mathrm{x} + 6}{\mathrm{x} + 1}\right)}^{\mathrm{x} + 4} = ?$

• e⁴

• e⁶

• e⁵

• e

The coordinates of the point P on the curve $\mathrm{x} = \mathrm{a}\left(\mathrm{\theta } + \sin \left(\mathrm{\theta }\right)\right), \mathrm{y} = \mathrm{a}\left(1 - \cos \left(\mathrm{\theta }\right)\right),$ where the tangent is inclined at an angle $\frac{\mathrm{\pi }}{4}$ to x-axis, are

• $\left[\mathrm{a}\left(\frac{\mathrm{\pi }}{4} - 1\right), \mathrm{a}\right]$

• $\left[\mathrm{a}\left(\frac{\mathrm{\pi }}{2} + 1\right), \mathrm{a}\right]$

• $\left[\mathrm{a}\left(\frac{\mathrm{\pi }}{2}\right), \mathrm{a}\right]$

• (a, a)

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = {\mathrm{xtan}}^{-1}\left(\mathrm{x}\right), \mathrm{then} \underset{\mathrm{x}\to 1}{\lim }\frac{\mathrm{f}\left(\mathrm{x}\right) - \mathrm{f}\left(1\right)}{\mathrm{x} - 1} = ?$

• $\frac{\mathrm{\pi } +\hspace{0.17em}3}{4}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi } + 1}{4}$

• $\frac{\mathrm{\pi } +\hspace{0.17em}2}{4}$

$\mathrm{If} \mathrm{g}\left(\mathrm{x}\right) = \frac{\mathrm{x}}{\left|\mathrm{x}\right|} \mathrm{for} \mathrm{x}> 2, \mathrm{then} \underset{\mathrm{x}\to 2}{\lim } \frac{\mathrm{g}\left(\mathrm{x}\right) - \mathrm{g}\left(2\right)}{\mathrm{x} - 2} = ?$

•  - 1

• 0

• 1/2

• 1

$\underset{\mathrm{x}\to \frac{\mathrm{\pi }}{2}}{\lim }\left(\frac{2\mathrm{x} - \mathrm{\pi }}{\cos \left(\mathrm{x}\right)}\right) = ?$

• 0

• 1/2

•  - 2

• 5

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

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

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

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

• 0