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?

Simply compute the left-hand and right-hand limits, denoted by $\lim_{x \to a^-}$ and $\lim_{x \to a^+}$, respectively. If the left-hand limit is not equal to the right-hand limit, then you have a discontinuity, and depending on their values, you can say exactly what type of discontinuity it is.
