I am having a hard time grasping the definition of a limit. I initially learned this loose definition of a limit:
$\lim\limits_{x \to a}f(x)=L$ iff $f(x)$ approaches L when $x$ approaches $a$ from both directions.
Then i learned the epsilon delta definition, which states the following:
$\lim\limits_{x \to a}f(x)=L$ iff for any $\epsilon > 0$ there exists some $\delta>0$ where if $0<|x-a|<\delta$ then $|f(x)-L]<\epsilon$
The way i think about this definition(please correct me if I'm wrong) is that for any given range $\epsilon$ around L, we can find some range $\delta$ around x such that the function evaluated at those points will be within $\epsilon$ of L. However i don't understand how that statement is the same thing as saying:
$\lim\limits_{x \to a}f(x)=L$ iff $f(x)$ approaches L when $x$ approaches $a$ from both directions.