Math Help - L'Hopital's Rule

1. L'Hopital's Rule

I have to use l'hopital's rule to find each limit:

1. lim x approaching 0 from the right of

(1-cos(sqrt x))/x

Since this is a 0/0 form I use l'hopitals rule. making it:

Lim x approaching 0 from the right of (sin(sqrt x))/(2 sqrt x).

Again this is a 0/0 form but if I just keep applying l'hopital's rule it always comes up with a 0/0 form. How do I solve this?

2. You just need to apply it once more.

(sin(sqrt x))/(2 sqrt x) -> 1/2x^(-1/2)sin(sqrt x) / x^(-1/2) simplifies to 1/2cos (sqrt x).

Forgive me if the details aren't right, but I'm pretty sure the idea is there.

3. I'd let $x=u^2$ which makes this much easier to work with

$\lim_{u\to 0^+}{1-\cos u\over u^2} ={1\over 2}\lim_{u\to 0^+}{\sin u\over u} ={1\over 2}$