TITLE: How to prove this limit does not exist
QUESTION [7 upvotes]: I'm a relative novice to epsilon-delta proofs. My professor assigned this practice problem and I'm having terrible trouble understanding the answer he gave. Moreover, I can't find a good account for a general strategy for how to do these sorts of proofs. I understand the epsilon delta definition, I understand what I'm supposed to do, but I need advice on strategy.
The actual problem.
Prove that the limit
$$ \lim_{x \rightarrow 2} \frac{x^3}{x-2} $$
does not exist.
So far I'm pretty stumped; I know I need to show that there is some $\epsilon$ st. such that x being arbitrarily close to 2 does not guarantee that f(x) is within epsilon of L, but that's all I've got.
Thanks.

REPLY [3 votes]: a limit exists iff 
$\exists a$ $\forall \epsilon$ $\exists \delta$ so that $|x-2|<\delta \implies |f(x)-a|<\epsilon$
To prove a limit does not exist, you need to prove the opposite proposition, i.e.
$\forall a$ $\exists \epsilon$ $\forall \delta$ so that $\exists x,|x-2|<\delta$ and $|f(x)-a|>\epsilon$
$2+\delta/2$ and $2-\delta/2$ are good candidates : they are close enough from 2, and their image by the function are really far apart, so one of them will be far away enough from your $\epsilon$