Proving $\lim _{x \to 0^-} \frac{1}{x}=-\infty$ How do I calculate the limit:
$$\lim _{x \to 0^-} \frac{1}{x}$$
The answer is clearly $-\infty$, but how do I prove it? Its clear that as x approaches 0 from the right side, $x$, becomes infinitely small, but negative, making the limit go to $-\infty$, but how do I prove this mathematically?

Notice $\displaystyle\frac{1}{x}$ is strictly decreasing on $(-1,0)$.
I claim that fo any positive integer $n$, if we choose $x$ close enough to $0$$, \displaystyle\frac{1}{x} < -n$. Hint: consider $x\in (-\frac{1}{n},0)$.
Do you see why it should follow that the limit is $-\infty$?
