Hi 315, you probably already know that I am not great at limits but i have looked at yours and i have a question.
$$\displaystyle \lim_{x\rightarrow 0^+}\left(sin\left(e^{-\frac{1}{x^2}}\right)\right)\\\\\\
=\displaystyle \lim_{x\rightarrow 0^+}\left(sin\left(\frac{1}{e^{\frac{1}{x^2}}}\right)\right)\\\\\\$$
I agree with you that this equal 0 but I don't understand why you added the middle step of =lim x tends to 0 of sine x. I mean i know that also tends to 0 but why have you put them together?
Is there something that I don't know about??