Removing holes

Theorem 2.1   If $${\lim_{x\to a} f(x)=L}$$ and $f(x)=g(x)$ except at $a$, then If $${\lim_{x\to a} g(x)=L}$$.

PROOF:

Given $\epsilon$ we can use $${\lim_{x\to a} f(x)=L}$$ to find $\delta_f$ such that if $0<\vert x-a\vert<\delta_f$ then $\vert f(x)-L\vert<\epsilon$. Since $0<\vert x-a\vert$ we know that $x\neq a$ so $f(x)=g(x)$. This tells us that $\vert g(x)-L\vert<\epsilon$. Thus $${\lim_{x\to a} g(x)=L}$$.

Commentary: This theorem allows us to make algebraic simplifications in limit calculations which change the domain, but don't change any values. Since this is a very common tactic in limit calculation it is desirable to have the theorem to justify those steps. It also helps expalin why we want the $0<\vert x-a\vert$ part of the definition.