As x goes to infinity, x-sin(sin(x)) and x-sin(x) are both between x-1 and x+1, so you can use the squeeze theorem. I think you can use L'Hôpital's rule for the follow-up.

I think you can also expand the numerator and denominator in Taylor series to get the limit as x goes to 0. To get the Taylor series for sin(sin(x)), you'll need to plug the series for sin(x) into itself - just figure out the first few terms.

- Hollywood