limit as x approaches zero from left and right equals positive and negative infinity

Question

How can we prove \begin{align} \lim_{x\to 0^+}\frac1x &= +\infty, \\ \lim_{x\to 0^-} \frac1x &= -\infty \end{align}

This seems really simple but I'm having trouble starting it

Answer 1

This is a very general advice, but it works every time:

If you do not know where to start, write down the definitions you know. Look at the definitions. Look especially at the definition of the thing you want to prove.

The definition of what you want to prove usually looks like

This thing you want to prove is true if and only if this other fancy expression is true.

Now, you just made a first step: you now know you need to prove the "other fancy expression". If you again do not know what to do, repeat the process with that.