Question

`lim_("h" -> 0) (1/("h"^2 sqrt(8 + "h")) - 1/(2"h"))` is equal to ____________.

Options

• `1/24`

• None of these

• `-1/48`

• `1/48`

Answer 1

`lim_("h" -> 0) (1/("h"^2 sqrt(8 + "h")) - 1/(2"h"))` is equal to `underline(-1/48)`.

Explanation:

`lim_("h" -> 0) (1/("h"^2 sqrt(8 + "h")) - 1/(2"h"))`

`=  lim_("h" -> 0)1/"h" {1/2 (1 + "h"/8)^(-1/2) - 1/2}`

`=  lim_("h" -> 0)1/"h".1/2 {1 - 1/3 . "h"/8 + ((-1/3)(-5/3))/2 "h"^2/64 + .... -1}`

`=  lim_("h" -> 0)1/(2"h") {-"h"/24 + 5/18 . "h"^2/64 + ....}`

`=  lim_("h"-> 0) 1/2 {-1/24 + 5/18 . "h"/64 + ...}`

`= -1/48`