Why can I put the limit sign in equations Why can I put the limit sign in both sides of an equation?
Is there a rigorous way to prove that if:
$f(x)=g(x)$
Then
$\lim \limits_{x\to x_0}f(x) =\lim \limits_{x\to x_0}g(x)$
Thanks.

Well, if $f(x)=g(x)$, then $f(x)=g(x)=f(x)$. 
Therefore, we have $\displaystyle \lim_{x \to x_0}[f(x)]=\lim_{x \to x_0}[g(x)]$, because we have an identical expression under a different name.
