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.

Assume that 
$$\forall x \in (x_0-\eta,x_0+\eta) \;\;F(x)=0$$
and let us prove that
$$L=\lim_{x\to x_0} F (x)=0.$$
suppose $L\ne 0$.
take $\epsilon=|L|/2$.
$\exists a>0  : \forall x\in (x_0-a,x_0+a) $ 
$$|F (x)-L|< | L|/2$$
but $F (x)=0$ then
$$|L|<|L|/2$$
which is not possible.
