Limits and Derivatives

If the normal to the curve y = f(x) at (3, 4) makes an angle $\frac{3\mathrm{\pi }}{4}$ with the positive x-axis, then f'(3) is equal to :

• - 1

• $\frac{3}{4}$

• 1

• - $\frac{3}{4}$

$\underset{\mathrm{h}\to 0}{\lim }\frac{\sqrt{\mathrm{x} +\hspace{0.17em}\mathrm{h}} - \sqrt{\mathrm{x}}}{\mathrm{h}}$ is equal to

• $\frac{1}{2\sqrt{\mathrm{x}}}$

• $\frac{1}{\sqrt{\mathrm{x}}}$

• $2\sqrt{\mathrm{x}}$

• $\sqrt{\mathrm{x}}$

$\underset{\mathrm{\theta }\to 0}{\lim }\frac{5\mathrm{\theta cos}\left(\mathrm{\theta }\right) - 2\sin \left(\mathrm{\theta }\right)}{3\mathrm{\theta } + \tan \left(\mathrm{\theta }\right)}$ is equal to

• 3/4

• - 3/4

• 0

• None of these

If cos(x + y) = ysin(x), then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $- \frac{\sin \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right) + \mathrm{ycos}\left(\mathrm{x}\right)}{\sin \left(\mathrm{x}\right) + \sin \left(\mathrm{x} + \mathrm{y}\right)}$

• $\frac{\sin \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right) + \mathrm{ycos}\left(\mathrm{x}\right)}{\sin \left(\mathrm{x}\right) + \sin \left(\mathrm{x} + \mathrm{y}\right)}$

• $\frac{- \sin \left(\mathrm{x} +\hspace{0.17em}\mathrm{y}\right) + \mathrm{ycos}\left(\mathrm{x}\right)}{\sin \left(\mathrm{x}\right) + \sin \left(\mathrm{x} + \mathrm{y}\right)}$

• None of the above

$\underset{\mathrm{x}\to 2^{+}}{\lim }\frac{\left|\mathrm{x} - 2\right|}{\mathrm{x} - 2}$ is equal to

• - 1

• 1

• 2

• - 2

$\mathrm{y} = {\tan }^{-1}\left(\frac{\sqrt{\left(1 + {\mathrm{x}}^2\right)} + \sqrt{\left(1 - {\mathrm{x}}^2\right)}}{\sqrt{\left(1 + {\mathrm{x}}^2\right)} - \sqrt{\left(1 - {\mathrm{x}}^2\right)}}\right), \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

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

• $- \frac{1}{2}$

• $- \frac{\mathrm{x}}{\sqrt{\left(1 - {\mathrm{x}}^4\right)}}$

• $\frac{\mathrm{x}\sqrt{1 + {\mathrm{x}}^2}}{1 - {\mathrm{x}}^4}$

If $\underset{\mathrm{x}\to \mathrm{a}}{\lim }\frac{{\mathrm{a}}^{\mathrm{x}} - {\mathrm{x}}^{\mathrm{a}}}{{\mathrm{x}}^{\mathrm{x}} - {\mathrm{a}}^{\mathrm{a}}} = - 1$, then

• a = 1

• a = 0

• a = e

• $\mathrm{a} = \frac{1}{\mathrm{e}}$

$\underset{\mathrm{x}\to 5}{\lim }\frac{\mathrm{x} - 5}{\left|\mathrm{x} - 5\right|}$ equals

• 2

• 0

• - 2

• None of these

$\underset{\mathrm{x}\to \frac{\mathrm{\pi }}{3}}{\lim }\frac{\sin \left(\frac{\mathrm{\pi }}{3} - \mathrm{x}\right)}{2\cos \left(\mathrm{x}\right) - 1}$ is equal to

• $\frac{1}{2}$

• $\frac{1}{\sqrt{3}}$

• $\frac{- 1}{\sqrt{3}}$

• $\frac{2}{\sqrt{3}}$

If 2^x + 2^y =2^{x + y}, then the value of $\frac{\mathrm{dy}}{\mathrm{dx}}$ at x = y = 1 is

• 0

• - 1

• 1

• 2