How do you know when you can substitute certain limits into others? 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).
 A: 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 $\lim_{x\to a} f(x)=f(a)$.

A: When you want to use approximations such as that $\sin x$ is approximately the same as $x$ for small $x$, then you should consider the limit of their quotient - her the well-known $\lim_{x\to 0}\frac{\sin x}x=1$. A very intuitive way of working with such approximations (though possibly not at your course level) is with the Big-Oh notation: $\sin x = x+O(x^3)$.
A: The statement "lim as x approaches $0$ of $f(x)/f(sin(x))=1$ is true. Why I was actually getting a wrong answer is because instead of replacing all of the x'es in f(x) with sin(x)'es, I was only replacing some of them. For example, an example would be the limit as $x$ approaches $0$ of $(arcsin(x)-x)/(x^3)$. I only tried to substitute $sin(x)$ for $x$ in the $arcsin$ function, therefore giving me $[arcsin(sin(x))-x]/x^3)=0$. However, the limit was $1/6$, and you do get $1/6$ if you replace $all$ of the $x'es$ with $sin(x)'es.$
