I know there are some limits where you can't to certain substitutions such as $\sin(x)=x$ as $x$ approaches $0$. How do you know when you can or can't do that? I wish I could give you an example because I saw one on this site a few days ago but I can't remember it. (By the way please keep the answers at a calc AB level).
Answer
It looks like you might have a slightly weird conception of what a limit is.
For a real valued function, the expression $\lim_{x\to a}f(x)$, when it exists, is a real number. So where you've written "$\sin(x)=x$ as $x$ goes to zero" you haven't properly expressed where the limit is taking place, and the two things on the sides of the equality are not equal at all, except at zero. It would be true to say that $\sin(0)=0$, but I think I agree with Cocopuffs that you probably meant to express $\lim_{x\to 0}\sin(x)=\sin(0)=0$.
This is the general rule (in the context of functions on the reals):
If $f:\Bbb R\to \Bbb R$ is continuous at $a$, then $\lim_{x\to a} f(x)=f(a)$.
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