Areas where the limit doesn't exist

Question

I am currently marking for a first year calculus class, and I think my prof made a mistake on the solution sheet for the most recent recitation.

The prof provided a cartesian coordinate system with a function $f(x)$, with vertical asymptotes at $x=-3, x=0$, and a horizantal asymptote at $y=1$.

At the asymptote at $x=0$, the function approaches $+ \infty$ from both sides
(ie $\lim_{x \to 0^+} = \lim_{x \to 0^-} = +\infty$).

Now, one of the questions asks to list all numbers $a$ which $\lim_{x \to a} f(x)$ does not exist. And my prof listed $x=0$ as a point where the limit doesn't exist. But wouldn't the $\lim_{x \to 0} = +\infty$, and thus shouldn't be not listed as a point where the limit doesn't exist?

Thanks for any help

Answer 1

Typically, we say that $\lim_{x\to a} f(x)$ exists if there is some real number $L$ such that for all $\varepsilon > 0$ we can find some $\delta > 0$ such that $$ |x-a| < \delta \implies |f(x) - L| < \varepsilon. $$ When we might write $$ \lim_{x\to a} f(x) = +\infty, $$ there is no real number $L$ that does this job. The limit doesn't exist in this case, but the behaviour of the function can be characterized. Specifically, for all $M > 0$, there exists some $\delta > 0$ such that $$ |x-a|<\delta \implies f(x) > M. $$ In other words, the function can be made arbitrarily large (and positive) by choosing $x$ close to $a$.