Speaking about left and right hand limits of 1/x as x tends to 0

Question

Let $x$ be a real variable. It’s valid to say,

As $x \rightarrow \infty$, $\frac{1}{x} \to 0 $

since $$\lim_{x \to \infty} \frac{1}{x} = 0$$

But is it also valid to say,

As $\frac{1}{x} \to 0$, $x \to \infty $

Since

$$\lim_{\frac{1}{x} \to 0^+} x = \infty$$

is different from

$$\lim_{\frac{1}{x} \to 0^-} x = -\infty$$

Answer 1

The first statement is correct, i.e. as $x \to \infty$, we have $\frac1x \to 0$.

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Conversely, if $\frac1x \to 0$ that does not imply that $x \to \infty$ since it could be $x \to -\infty$, as you indeed point out. You could, however, say that as $\frac1x \to 0$ we have $|x| \to \infty$, or as $\frac1x \to 0^+$, we have $x \to \infty$.

Answer 2

No because $1/x$ can go to zero in more than one case. $x$ can go to plus infinity or minus infinity or bounces up and down and its absolute value goes to infinity.

What we can say for sure is that if $1/x$ goes to zero then $|x|$ goes to infinity.