I use Stewart's (Calculus, 8e) terminology. Infinite limits do not exist. For example we can write $$\lim_{x \rightarrow 0} \frac{1}{x^2} = \infty, $$ but at the same time say that $$\lim_{x \rightarrow 0} \frac{1}{x^2}$$ does not exist. Or at least this is what Stewart (89ff) insists.
My question is this: Do limits at infinity exist? For example we can write $$\lim_{x \rightarrow \infty} \frac{1}{x^2} = 0.$$ But in this case, should we say that $$\lim_{x \rightarrow \infty} \frac{1}{x^2}$$ exists or does not exist? Stewart (126ff) doesn't seem to explicitly address this, which is why I'm asking.
(Related: Why does an infinite limit not exist? — here the reference is again Stewart, but there seems to be some confusion of the terms "infinite limit" and "limit at infinity".)
Best Answer
Any statement (or equation) involving the symbol $\infty$ has a precise meaning not by default (or via knowledge of primary school level math) but via a special definition to interpret such statements. So if you write $$\lim_{x \to 0}\frac{1}{x^{2}} = \infty$$ then it does not mean that the symbol $$\lim_{x \to 0}\frac{1}{x^{2}}$$ is some specific thing and the symbol $\infty$ is another specific thing and both are equal. Rather this equation has a special meaning given by a specific definition which is as follows:
Given any real number $N > 0$, there is a real number $\delta > 0$ such that $$\frac{1}{x^{2}} > N$$ whenever $0 < |x| < \delta$.
Any textbook must define the precise meaning of phrases containing the symbol $\infty$ (and equations containing the symbol $\infty$) before writing such phrases (or equation). If this is not done then the textbook author is guilty of a common crime called "intellectual dishonesty".
On the other hand there are many conventions about the existence of a limit. Some authors prefer to say that a limit exists only when it is finite (I prefer this approach). Some define infinite limits also as a case of existence of the limit. Both the approaches are rigorous and without any fault. It is a matter of taste and individual preference of authors. For a student it is of utmost importance to follow the convention prescribed by his teacher/instructor.
The other limit $$\lim_{x \to \infty}\frac{1}{x^{2}} = 0$$ exists in both the conventions and I don't see any problem here.