The standard $\epsilon-\delta$ definitions do not apply to limits with infinity because $\infty\pm\epsilon$ has no meaning.
Instead, the definition becomes something like this for real valued functions:
$\lim_{x\rightarrow c}f(x)=\infty$ means that for all $n\in\mathbb{N}$, there exists $\delta>0$ such that for all $x$, if $|x-c|<\delta$, then $f(x)>n$.
This encapsulates what it means for the function to "get close to infinity."
As an exercise, you can write what it means for $f(x)$ to have limit $-\infty$ as $x$ approaches $c$.
For your $x^3/|x|$ problem, looking at the left and right limits helps you get rid of the absolute value signs. If $x\rightarrow 0^-$, then $x$ is a small negative number approaching zero, and your function is $x^3/|x|=x^3/(-x)=-x^2$. On the other hand, for $x\rightarrow 0^+$, it is a shrinking positive number, so $x^3/|x|=x^3/x=x^2$.
The two-sided limit exists iff you find the left and the right limits exist, and they both are the same number.