Limits and Derivatives

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

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

• - $\frac{3}{4}$