Answer
See the proof below.
Work Step by Step
To prove $\lim\limits_{x \to 3}(3x-7)=2$, we let $ c=3, f(x)=3x-7, L=2$ as used in the definition of limits.
For any given small value of $\epsilon\gt0$, we need to find a value of $\delta $ so that for all $ x, 0\lt |x-c|\lt\delta $, we should have $|f(x)-L|\lt\epsilon $.
or $|3x-7-2|\lt\epsilon, |3(x-3)|\lt\epsilon $.
Let $\delta=\epsilon/3$, for all $ x, 0\lt|x-3|\lt\delta=\epsilon/3$, we have:
$|f(x)-L|=|(3x-7)-2|=3|x-3|\lt\epsilon $ which proves the limit statement.
