
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
(1cos(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?

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.

I'd let $\displaystyle x=u^2$ which makes this much easier to work with
$\displaystyle \lim_{u\to 0^+}{1\cos u\over u^2} ={1\over 2}\lim_{u\to 0^+}{\sin u\over u} ={1\over 2}$