This is Ana's Definition of the Derivative.
For limits to infinity, we can compare the numerator with the denominator. As the numerator approaches infinity, y = sin²x oscillates between −1 and +1... not very big. As the denominator approaches infinity, y = x also approaches infinity. So the denominator is MUCH larger than the numerator.
See the hint in part (a).
Recall that the number e is about 2.718. It's larger than 2 and smaller than 3. Use that to compare the power of the numerator and denominator.