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?
 A: 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...
A: 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.
A: Jump discontinuity for $f(x)$ at $x = a$.
According to the definition of "jump discontinuity" in your textbook (which I must say as a disclaimer runs counter to what I've been taught), this can be characterized as follows:


*

*$\displaystyle\lim_{x \to a^-} f(x)$ and $\displaystyle\lim_{x \to a^+} f(x)$ are both real numbers and are equal to each other, but $f(a)$ is a different real number.



Point discontinuity (aka removable discontinuity) for $f(x)$ at $x=a$.
Same disclaimer as above.


*

*$\displaystyle\lim_{x \to a^-} f(x)$ and $\displaystyle\lim_{x \to a^+} f(x)$ are both real numbers and are equal to each other, and $f(x)$ is not defined at $x=a$.



Vertical asymptote for $f(x)$ at $x=a$:


*

*$\displaystyle\lim_{x \to a^-} f(x) = \pm \infty$ and $\displaystyle\lim_{x \to a^+} f(x) = \pm \infty$.  Note that they don't have to both be $+\infty$ or $-\infty$.  One can be $+\infty$ and the other can be $-\infty$.



Jump discontinuity (step) for $f(x)$ at $x=a$.


*

*$\displaystyle\lim_{x \to a^-} f(x)$ and $\displaystyle\lim_{x \to a^+} f(x)$ are both real numbers and are not equal to each other, and $f(a)$ is equal to one of these limits, or potentially $f(a)$ could be a different value entirely, or not defined at all.

