Limits and Derivatives

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

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

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

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

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

$\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

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

• $\infty$

• $\frac{1}{2}$

• 4

• 1

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

• $- \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.$

If f(x) = $$\begin{gathered} \left\{\frac{\sin \left(1 + \left[\mathrm{x}\right]\right)}{\left[\mathrm{x}\right]}, \mathrm{for} \left[\mathrm{x}\right] \ne 0 \\ 0 , \mathrm{for} \left[\mathrm{x}\right] = 0\right\} \end{gathered}$$

where [x] denotes the greatest integer not exceeding x, then $\underset{\mathrm{x}\to 0^{-}}{\lim }\mathrm{f}\left(\mathrm{x}\right)$ is equal to

• - 1

• 0

• 1

• 2

If f(x) = $$\begin{gathered} \left\{\mathrm{x} - 5, \mathrm{for} \mathrm{x} \le 1 \\ 4{\mathrm{x}}^2 - 9, \mathrm{for} 1 < \mathrm{x} < 2 \\ 3\mathrm{x} + 4, \mathrm{for} \mathrm{x} \ge 2\right\} \end{gathered}$$

then f'(2⁺) is equal to

• 0

• 2

• 3

• 4

If 2x² - 3xy + y² + x + 2y - 8 = 0, then : $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{3\mathrm{y} - 4\mathrm{x} - 1}{2\mathrm{y} - 3\mathrm{x} + 2}$

• $\frac{3\mathrm{y} - 4\mathrm{x} + 1}{2\mathrm{y} + 3\mathrm{x} + 2}$

• $\frac{3\mathrm{y} - 4\mathrm{x} + 1}{2\mathrm{y} - 3\mathrm{x} - 2}$

• $\frac{3\mathrm{y} - 4\mathrm{x} + 1}{2\mathrm{y} + 3\mathrm{x} + 2}$

$$\begin{gathered} \mathrm{If} \mathrm{z} = \log \left(\tan \left(\mathrm{x}\right) + \tan \left(\mathrm{y}\right)\right), \mathrm{then} \\ \sin \left(2\mathrm{x}\right)\frac{\partial \mathrm{z}}{\partial \mathrm{x}} + \sin \left(2\mathrm{y}\right)\frac{\partial \mathrm{z}}{\partial \mathrm{y}} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• 1

• 2

• 3

• 4

$\underset{\mathrm{x}\to 0}{\lim } \frac{\left(1 - {\mathrm{e}}^{\mathrm{x}}\right)\sin \left(\mathrm{x}\right)}{{\mathrm{x}}^2 + {\mathrm{x}}^3} = ?$

• - 1

• 0

• 1

• 2

$\mathrm{If} \mathrm{x} = \mathrm{a}\left\{\cos \left(\mathrm{\theta }\right) + \log \left(\tan \left(\frac{\mathrm{\theta }}{2}\right)\right)\right\} \mathrm{and} \mathrm{y} = \mathrm{asin}\left(\mathrm{\theta }\right), \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}} = ?$

• $\cot \left(\mathrm{\theta }\right)$

• $\tan \left(\mathrm{\theta }\right)$

• $\sin \left(\mathrm{\theta }\right)$

• $\cos \left(\mathrm{\theta }\right)$