I do not understand why limits equal to infinity can in certain circumstances exist while in others they do not.
For instance, the limit does not exist for the following:
However, I believe the limit is equal to $-\infty$ in this case: 
Why??? Why is limit 2 a special case? I don't understand. Is it because it can be simplified using long division?
Help on this would be greatly appreciated.
Note that "the limit is equal to $-\infty$" is not a precise statement, or rather that the function approaching $-\infty$ in the tail does NOT mean the limit exists - for the limit to exist it can only be a real number.
The limit does not exist in either example above. While it's still not absolutely precise it is common to say "approaches infinity" to mean grows in an unbounded fashion - there are other ways for a limit to not exist, e.g. a sequence that bounces back and forth between two values.
The way to evaluate these quickly (without formal proof, although this reasoning can be justified) is just to compare highest powers in the numerator and denominator, and constants can be ignored (except in the case where the highest powers agree).
The first example has the same tail behavior as $\frac x{x^{2/3}} = \sqrt[3]{x}$ which approaches $+\infty$ and the second behaves like $\frac {x^2}x = x$ which approaches $-\infty$.