find the limit as x-> inf for f(x)= cos(x)/sqrt(x)

so, I can tell it's going to be 0, but as a proof i'm not sure what to say

I started by changing it to sqrt(x)*cos(x)/x <= sqrt(x)/(x) = 1/sqrt(x)

Which obviously goes to zerp as x goes to infinity...but what about a lower bound?

Another problem is lim x -> 0 for f(x) = sin^2(sin(x))

All i've really got is that it is bounded below by 0 and above by 1, not really sure if a epsilon delta proof would be better...