Limits and Derivatives

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

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

Use the formula $\underset{\mathrm{x}\to 0}{\lim }\frac{{\mathrm{a}}^{\mathrm{x}} - 1}{\mathrm{x}} = {\log }_{\mathrm{e}}\left(\mathrm{a}\right) \mathrm{to} \mathrm{compute} \underset{\mathrm{x}\to 0}{\lim }\frac{2^{\mathrm{x}} - 1}{\sqrt{1 + \mathrm{x}}{\displaystyle }{\displaystyle -}{\displaystyle }{\displaystyle 1}}$

The value of $\underset{\mathrm{x}\to 0}{\lim }\frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right) - 1}{{\mathrm{x}}^2}$

• 1

• $\frac{1}{2}$

• $\frac{- 1}{2}$

• 0

The value of $\underset{\mathrm{x}\to 0}{\lim }{\left(\frac{1 + 5{\mathrm{x}}^2}{1 + 3{\mathrm{x}}^2}\right)}^{\frac{1}{{\mathrm{x}}^2}}$

• e²

• e

• $\frac{1}{\mathrm{e}}$

• $\frac{1}{{\mathrm{e}}^2}$

If f(5)=7 and f'(5)=7 then $\underset{\mathrm{x}\to 5}{\lim }\frac{\mathrm{xf}\left(5\right) - 5\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{x} - 5}$ is given by

• 35

• - 35

• 28

• - 28

The value of f(0) so that the function $\frac{1 - \cos \left(1 - \cos \left(\mathrm{x}\right)\right)}{{\mathrm{x}}^4}$ is continuous everywhere is

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{6}$

• $\frac{1}{8}$

$\underset{\mathrm{x}\to 0}{\lim }\frac{\sin \left|\mathrm{x}\right|}{\mathrm{x}}$ is equal to 

• 1

• 0

• positive infinity

• does not exist