In a function such that $f(x)=\frac{1}{x^2}$ would the limit as $x$ approaches zero be infinity or would it not exist?

Question

If you have a function such as $f(x)=\frac{1}{x^2}$ , would the limit be infinity or would it not exist. A textbook I had said it was infinity but I feel like it does not exist because infinity is not a specific numeric value. So what is the limit for such an equation?

Answer 1

Infinity is not really a number, at least not in standard analysis. The limit does not exist.

Answer 2

It depends on the author's persuasion. Really, saying something like $\lim_{x \to 0} \frac{1}{x^2} = +\infty$ is just short-hand for "the limit does not exist but here's why." The function consistently blows up from either side of $0$, so that's how you report it. But, it definitely does not exist, even though you write it as "equal" to something.

This explains why the same author may write "$\lim_{x \to 0} \frac{1}{x}$ does not exist." Here, the function exhibits different/inconsistent behavior from each side: one blows up, one blows down. So, you can not report that it does something consistent around $0$ regardless of side, so you have no choice but to use the words.

To answer the question in your title, the answer is unfortunately "both." The limit does not exist because it approaches infinity.