Question

How do you find the Limit of `(sin^3 x )/ (sin x - tan x)` as x approaches 0?

Answer 1

`-2`

Explanation:

`(sin^3 x )/ (sin x - tan x)=(sin x cdot sin^2x)/(sinx(1-1/cosx))=sin^2x/(1-1/cosx)`
`=((1-cos^2x)cosx)/(cosx-1)=-((1-cosx)(1+cosx)cosx)/(1-cosx)`

`=-(1+cosx)cosx`

then

`lim_{x->0}(sin^3 x )/ (sin x - tan x)=lim_{x->0}(-(1+cosx)cosx) = -2`