Analysis question- please help.

Could somebody please help me to do the following questions? I am really stuck.

1.a. A function f, from the reals to the reals is continuous at 0 with f(0)=0 and another function g from the reals to the reals is bounded then the product fg is continuous at 0.

I think this may be true but don't know how to start a proof, any ideas?

b. If for f from [-1,1] to the reals, sin(f(x)) is continuous (over the reals) then f must be continuous.

I think f(x)=1/x (restricted to [-1,1]) may be a counterexample to this but one problem is it is not defined at 0 which is in [-1,1], is it ok to use it anyway.

I think this may be true but don't know how to start a proof, any ideas?

c. If for f from [-1,1] to the reals f(sinx) is continuous on R then f is continuous on R

2. Find without proof the following limits.

a. cosx/sinx -1/x as x tends to 0 from below.

b. x^x as x tends to 0 from above.

I don't see how to find these limits (short of plugging small numbers into a calculator and seing what seems to be happening). As I won't have a calculator in my exam I was wondering, is there another way?