Is this sufficiently logical to commit to memory? Any suggested edits?
If $x$ is within $\delta$ of $c$,
so $f$ is in $\varepsilon$ of $T;$
work $f$ minus $T$ to $x$ minus $c$;
choose $\delta$ as $\varepsilon$, worked as need be;
then the limit of $f$ as $x$ goes to $c$
is proven to be equal to $T$.