Given a function, how can one tell if it doesn't have a limit at $x=a$ due to a discontinuity? For example, if you have the $$\lim_{x \to 2} \frac{1}{x-2},$$ the limits approaching from the positive and negative are different. You can tell because the $x-2$ becomes $0$ and the entire binomial is raised to an odd power.
How would you tell if  a function has a jump discontinuity, point discontinuity, jump discontinuity (step) or vertical asymptote such that it does not have a valid limit at a given point?

You have to look at the one-sided limits
$$
\lim_{x \to a^-} f(x) \quad \text{and} \quad \lim_{x \to a^+} f(x).
$$
The 2-sided limit exists iff both one-sided limits exist and are equal to each other, and $f(a)$ also has that value.
If 2-sided limits exist and are equal, but $f(a)$ has a different value, you have a point discontinuity. When the limits are not equal (like in greatest integer function) it is a jump discontinuity. When limits are infinite and different, (i.e. one is $+\infty$ and one is $-\infty$) you get an asymptote in different directions from each side...
