Question

`lim_(x->0)(sqrt(cosx)-root(3)(cosx))/sin^2x`?

Answer 1

`-1/12`

Explanation:

Let `cos x = t`
`lim_{x -> 0}(sqrt(cosx)-root(3)(cosx))/sin^2x = lim_{t-> 1}{t^(1/2)-t^(1/3)}/(1-t^2)`

This is of the form `0/0`, so we can use l'Hospital's rule:

`lim_{t-> 1}{t^(1/2)-t^(1/3)}/(1-t^2) = lim_{t-> 1}{d/dt(t^(1/2)-t^(1/3))}/{d/dt(1-t^2)} = lim_{t-> 1}{1/2t^(-1/2)-1/3t^(-2/3)}/(-2t)`
`qquad = (1/2-1/3)/(-2) = -1/12`