My prof said there was three ways, so far I've found two:
When the limit goes up to infinity, and when the left and right limits don't match. What are the other ways a limit cannot exist?
There are indeed three ways in which the $\displaystyle \lim_{x\to a}f(x)$ cannot exist.
1.$\displaystyle f(x)$ is not well defined inthe neighbourhood of $\displaystyle x=a$.
For example $\displaystyle \lim_{x\to 1}\sec^{-1}(\sin x)$ does not exist
since $\displaystyle \sec^{-1}(\sin x)$ is not defined in the neighbourhood
of $\displaystyle x=1$ though it is defined at $\displaystyle x=1$.The same
is true for $\displaystyle \lim_{x\to 1}\sin^{-1}(\frac{1+x^2}{2x})$.
2.$\displaystyle f(x)$ does not display tendency to approach a fixed numerical value.
For e.g. it is not known as to what value
$\displaystyle \lim_{x\to 0}\sin (\frac{1}{x}) $will assume though we are
sure that it is a finite quantity on the interval $\displaystyle [-1,1].$
Similarly, $\displaystyle f(x)=\begin{cases}0,\mbox { if } x\in irrationals\\1,\mbox{ if } x\in rationals\end{cases}$
Here, $\displaystyle f(3) = 1$.But if you try to evaluate $\displaystyle \lim_{x\to 3}f(x)$ we will not be able to decide to what value should f(x)
assume on the interval $\displaystyle [3-h,3+h]\notin {3}$.
3.Of course ,the last condition being Left hand limit$\displaystyle \neq $Right hand limit