Limits and Derivatives

$\underset{\mathrm{x}\to \mathrm{\alpha }}{\lim }{\left(\tan \left(\mathrm{x}\right)\cot \left(\mathrm{x}\right)\right)}^{\frac{1}{\mathrm{x} - \mathrm{\alpha }}}$ is equal to

• $2\csc \left(2\mathrm{\alpha }\right)$

• $\frac{1}{2}\sin \left(2\mathrm{\alpha }\right)$

• $- 2\csc \left(2\mathrm{\alpha }\right)$

• None of these

The value of $\underset{\mathrm{x}\to \infty }{\lim }\frac{{\left(2^{{\mathrm{x}}^{\mathrm{n}}}\right)}^{{\displaystyle {\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{e}$}\right.}^{\mathrm{x}} }}- {\left(3^{{\mathrm{x}}^{\mathrm{n}}}\right)}^{{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{e}$}\right.}^{\mathrm{x}} }}{{\mathrm{x}}^{\mathrm{n}}}$ (where, x $\in$ N) is

• 0

• $\mathrm{n} \log \left(\mathrm{n}\left(\frac{2}{3}\right)\right)$

• $\log \left(\left(\frac{2}{3}\right)\right)$

• not defined

$\underset{\mathrm{x}\to \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\lim }\frac{\tan \left(\mathrm{x}\right) - 1}{\cos \left(2\mathrm{x}\right)}$ is equal to

• 1

• 0

• - 2

• - 1

Which of the following inequality is true for x > 0?

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

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

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

• $\frac{\mathrm{x}}{1 + \mathrm{x}} < \log \left(1 +\hspace{0.17em}\mathrm{x}\right) < \mathrm{x}$

The value of $\underset{\mathrm{x}\to \infty }{\lim }{\left(\frac{\mathrm{\pi }}{2} - {\tan }^{-1}\left(\mathrm{x}\right)\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.}$ is

• 0

• 1

• - 1

• e

Find the value of the limit $\underset{\mathrm{x}\to 0}{\lim }\frac{\sqrt{1 - \cos \left(\mathrm{x}\right)}}{\mathrm{x}}$

• 0

• 1

• $\sqrt{2}$

• does not exist

The values of constants a and b so that

$\underset{\mathrm{x}\to \infty }{\lim }\left(\frac{{\mathrm{x}}^2 + 1}{\mathrm{x} + 1} - \mathrm{ax} - \mathrm{b}\right) = 0$ are

• a = 0, b = 0

• a = 1, b = - 1

• a = - 1, b = 1

• a = 2, b = - 1

$\underset{\mathrm{n}\to \infty }{\lim }\left[\frac{1}{1 . 2} + \frac{1}{2 . 3} + \frac{1}{3 . 4} + ... + \frac{1}{\mathrm{n}\left(\mathrm{n} + 1\right)}\right]$ is equal to

• 1

• - 1

• 0

• None of these

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

• e

• e²

• e³

• e⁵

If g(x) is a polynomial satisfying g(x) g(y) = g(x) + g(y) + g(xy) - 2 for all real x and y and g(2) = 5, then $\underset{\mathrm{x}\to 3}{\lim }$ g(x) is

• 9

• 10

• 25

• 20