Answer
$\lim\limits_{x \to -\infty}\frac{1}{x} = 0$
Work Step by Step
Let $f(x) =\frac{1}{x}$
This function is defined on the interval $(-\infty, 0)$
Let $\epsilon \gt 0$ be given.
Let $N = -\frac{1}{\epsilon}$
Suppose that $x \lt N$
Then:
$\vert \frac{1}{x} - 0\vert \lt \vert \frac{1}{N}\vert = \vert \frac{1}{(-\frac{1}{\epsilon})} \vert = \epsilon$
Therefore, $\lim\limits_{x \to -\infty}\frac{1}{x} = 0$
