Answer
$\lim\limits_{x \to 2} \frac{1}{x} = \frac{1}{2}$
Work Step by Step
Let $\epsilon \gt 0$ be given.
Let $\delta = min\{1, \epsilon\}$
Suppose that $\vert x-2 \vert \lt \delta$
Note that $1 \lt x \lt 3$
Then:
$\vert \frac{1}{x}-\frac{1}{2}\vert = \vert \frac{2-x}{2x} \vert = \vert \frac{1}{2x} \vert \cdot \vert 2-x \vert \lt (\frac{1}{2})(\delta) \leq (\frac{1}{2})(\epsilon) \lt \epsilon$
Therefore, $\lim\limits_{x \to 2} \frac{1}{x} = \frac{1}{2}$
