Limits to infinity?

Question

As a part of homework, I was asked

What does $\lim_{x\to a} f(x)=\infty$ mean?

In an earlier calculus class I was taught that in order for $L=\lim_{x\to a}f(x)$ to exist, we need that $L=\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)$. In an explicit(but extremely non-rigorous) example the teacher explained with a graphic in which the two two sides of a vertical asymptote tended to $\infty$, the limit didn't exist because we couldn't equate both lateral limits, since we cannot compare two infinities. Is this right? If so, I would answer "It doesn't make sense, since we would need that both lateral limits at $a$...equated to the same infinity?" If it is not right and that does make some sense, may you clarify this point to me about that definition?

Answer 1

$\lim_{x\to a}f(x)\to\infty$ means that as you go closer and closer to $a$, the value of $f(x)$ grows arbitrarily large. Now, if you approach $-\infty$ as you go closer to $a$ from the left (or right), and if you approach $+\infty$ as you go closer to $a$ from the right (or left), then the limit does not exist. The reason is that the limit has two values at the same point. To be mathematically precise, since we know that the left- and right-hand limits are the same if and only if the ordinary limit exists, and that the left- and right-hand limits are not the same, $\lim_{x\to a}f(x)$ is undefined. I believe that this is what you mean by "two sides of a vertical asymptote tended to $∞$."

Answer 2

There is still some difference of feeling on how to talk about these. In the traditional discussion of limits, a limit should be a finite number that is "approached arbitrarily closely" by the function (made rigorous by $ \ \epsilon - \delta \ $ proofs, or the like). So when a function does not approach a finite value as $ \ x \ \rightarrow \ a \ , $ we say the limit "does not exist".

But there are various ways this can happen. The oxymoronic phrase "infinite limit" is introduced in some texts to indicate that a function grows "without limit" to the same signed infinity for $ \ x \ $ approaching $ \ a \ $ "from both sides". Hence, these authors will write $ \ \lim_{x \rightarrow 0} \ \frac{1}{x^2} \ = \ +\infty \ , $ for example; they would still write $ \ \lim_{x \rightarrow 0} \ \frac{1}{x} \ $ DNE .