Explain this consequence of continuity A consequence of continuity is the following fact:
if $f(x)$ is continuous at $x=b$ and $\lim\limits_{x \to a} g(x)=b$, then,
$\lim\limits_{x \to a} f(g(x)) = f(\lim\limits_{x \to a}g(x))$  
with this fact we can solve the following:
$\lim\limits_{x \to 0} e^{\sin x}= e^{\lim\limits_{x \to a}\sin x} = e^0 = 1$  
so the fact is saying that in this case $f(\lim\limits_{x \to a}g(x)) = f(b).$
I don't understand how this fact helps solve the problem, is it because $\exp$ and $\sin$ are both continuous everywhere, and if they're continuous at the same places, you can use this fact?  
Please elaborate. 
thanks!

This solve your problem because they are both continuous function every where, you can pass the limit like this $\lim_{x\to x_0}f(g(x))=f(\lim_{x\to x_0}g(x))$ two times.
