Question

`lim_("x"-> pi) (1 + "cos"^2 "x")/("x" - pi)^2` is equal to ____________.

Options

• `3/2`

• `1/2`

• `1/3`

• None of these

Answer 1

`lim_("x"-> pi) (1 + "cos"^2 "x")/("x" - pi)^2` is equal to `underline(3/2)`.

Explanation:

`=  lim_("x"-> pi) (1 + "cos"^2 "x")/("x" - pi)^2`

`=  lim_("h" -> 0) (1 + "cos"^2 (pi + "h"))/("h")^2`

`=  lim_("h" -> 0) (1 - "cos"^2 "h")/("h")^2`

`=  lim_("h" -> 0) (1 - "cos h")/("h")^2 . lim_("h" -> 0) (1 + "cos h" + "cos"^2 "h")`

`= 1/2 (1 + 1 + 1) = 3/2`