The $e\psilon-\delta$ definition states that the limit of $f(x) = L$ as $x$ approaches $a$ if for all $\epsilon > 0$, there exists a $\delta > 0$, such that if $|x-a| < \delta$, then $|f(x) - L| < \epsilon$. The applet below courtesy of Sylvain Berube, shows its geometric interpretation.