Answer
Please see below.
Work Step by Step
We want to prove that $$\lim_{x \to 3^+}\frac{1}{x-3}=\infty$$ by using $\epsilon - \delta$ definition; that is, we must show that for each $M >0$, there exists a $\delta >0$ such that $\frac{1}{x-3} > M$ whenever $33$, $\frac{1}{x-3} >M$ if and only if $x-30$ we conclude that$$3 \frac{1}{\frac{1}{M}}=M .$$ Done.
