# 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?

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?

• Please read this tutorial on how to typeset mathematics on this site. – N. F. Taussig Sep 11 '18 at 2:18

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.