# Precise Def'n of Limit Proving the Sum Rule

Alrighty-then

I am going over a proof in Thomas' Calculus and I am not understanding a step.

Given that $\lim_{x\rightarrow c}f(x)=L \text{ and } \lim_{x\rightarrow c}g(x)=M$ prove that $\lim_{x\rightarrow c}(f(x)+g(x))=L+M$

Solution: He uses the triangle inequality to break up the summation

$|f(x)+g(x))-(L+M)|\le|f(x)-L|+|g(x)-M|$

Then he says:

"Since $\lim_{x\rightarrow c}f(x)=L$, there exists some number $\delta_1>0$ such that for all x

$0<|x-c|<\delta_1\Rightarrow|f(x)-L|<\epsilon/2$ "

I get lost there. Why $\epsilon/2$?

Thanks!

PS: Please don't yell at me Dick

Dick