To prove that $ \lim_{x\to 0^-} \frac{1}{x} = -\infty$, how do we choose $\delta$?

Question

Could someone please explain to me what delta would work for this proof? $$ \lim_{x\to 0^-} \frac{1}{x} = -\infty. $$

My professor used $\delta = -1/\beta$, but I don't understand how that makes sense.

Answer 1

Hint. $$ \frac{1}{-\frac{1}{\beta}} =\ ? $$

Answer 2

Presumably, $\beta$ appears in the definition of an infinite limit that you were given. It might have been stated as:

$\lim\limits_{x\to a-} f(x) = -\infty$ if for every $\beta<0$ there exists $\delta>0$ such that $f(x)<\beta$ whenever $-\delta<x-a<0$.

In your situation, you look for $\delta $ such that $$ \frac{1}{x}<\beta \quad\text{whenever } -\delta<x<0 $$ Rearrange the desired inequality (on the left) to isolate $x$ on one side (be extra careful with negative quantities). You will get $x>-1/\beta$. This is the reason for choosing $\delta=-1/\beta$.