The nature of the $\epsilon-\delta$ definition is that it takes in a function $f(x)$ a limit point $a$, and a limit value $L$, and provides a definition for $\lim_{x\to a} f(x) = L$. Thus the nature of the beast is not one of calculation - it requires even a proof to show that if $\lim_{x\to a} f(x)=L_1$ and $\lim_{x\to a} f(x)=L_2$ then $L_1=L_2$, because the definition does not assert that only one limit can exist.
It is best to realize that the $\epsilon$ definition of limits was arrived at as a formalism of something that all mathematicians understood. They had been using limits and continuity for centuries before they came up with this definition. The definition just finally gave a strong form to validate a limit value.