Mathematics : Limits and Derivatives

The limit of $\mathrm{xsin}\left({\mathrm{e}}^{\frac{1}{\mathrm{x}}}\right)$ as $\mathrm{x} \to 0$

• is equal to 0

• is equal to 1

• is equal to $\frac{\mathrm{e}}{2}$

• does not exist

The limits of $\sum _{\mathrm{n} = 1}^{1000}{\left(- 1\right)}^{\mathrm{n}}{\mathrm{x}}^{\mathrm{n}} \mathrm{as} \mathrm{x} \to \infty$

• does not exist

• exists and equals to 0

• exists and approaches to $+ \infty$

• exists and approaches to $- \infty$

If f(x) = e^x(x - 2)², then

• f is increasing in (- $\infty$, 0) and (2, $\infty$) and decreasing in (0, 2).

• f is increasing in (- $\infty , 0$) and decreasing in ($0, \infty$)

• f is increasing in (2, $\infty$) and decreasing in (- $\infty$, 0).

• f is increasing in (0, 2) and decreasing in ($- \infty , 0$) and (2, $\infty$)

$\underset{\mathrm{x}\to 0}{\lim }\frac{{\mathrm{\pi }}^{\mathrm{x}} - 1}{\sqrt{1 + \mathrm{x}} - 1}$

• does not exist

• equals log_e(${\mathrm{\pi }}^2$)

• equals 1

• lies between 10 and 11

The value of $\underset{\mathrm{n}\to \infty }{\lim } \frac{{\left(\mathrm{n}!\right)}^{{\displaystyle \frac{1}{\mathrm{n}}}}}{\mathrm{n}}$

• 1

• $\frac{1}{{\mathrm{e}}^2}$

• $\frac{1}{2\mathrm{e}}$

• $\frac{1}{\mathrm{e}}$

The approximate value of $\sqrt[5]{33}$ correct to 4 decimal places is

• 2.0000

• 2.1001

• 2.0125

• 2.0500

The value of $\underset{\mathrm{n}\to \infty }{\lim }\sum _{\mathrm{r} = 1}^{\mathrm{n}}\frac{{\mathrm{r}}^3}{{\mathrm{r}}^4 + {\mathrm{n}}^4}$ is

• $\frac{1}{2}{\log }_{\mathrm{e}}\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$

• $\frac{1}{4}{\log }_{\mathrm{e}}\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$

• $\frac{1}{4}{\log }_{\mathrm{e}}\left(2\right)$

• $\frac{1}{2}{\log }_{\mathrm{e}}\left(2\right)$

The value of $\underset{\mathrm{x}\to 1}{\lim }\frac{\mathrm{x} + {\mathrm{x}}^2 + ... + {\mathrm{x}}^{\mathrm{n}} - \mathrm{n}}{\mathrm{x} - 1}$

• n

• $\frac{\mathrm{n} + 1}{2}$

• $\frac{\mathrm{n}\left(\mathrm{n} + 1\right)}{2}$

• $\frac{\mathrm{n}\left(\mathrm{n} - 1\right)}{2}$

$\underset{\mathrm{x}\to 0}{\lim }\frac{\sin \left({\mathrm{\pi sin}}^2\left(\mathrm{x}\right)\right)}{{\mathrm{x}}^2}$ is equal to

• ${\mathrm{\pi }}^2$

• 3$\mathrm{\pi }$

• 2$\mathrm{\pi }$

• $\mathrm{\pi }$

If N = n!(n $\left(\mathrm{n} \in \mathrm{N}, \mathrm{n} > 2\right)$, then find

$\underset{\mathrm{N}\to \infty }{\lim }\left[{\left({\log }_2\left(\mathrm{N}\right)\right)}^{- 1} + \left({\log }_3{\left(\mathrm{N}\right)}^{- 1}\right) + ... + {\log }_{\mathrm{n}}{\left(\mathrm{N}\right)}^{- 1}\right]$