Determine whether this function is continuous at x = 2. If it is discontinuous, is the discontinuity removable?

We can use the Definition of Continuity:

Is the function defined at x = 2? As shown in the graph, and from the fact that the denominator is 0 at this point, we conclude that the answer is no.

Does the function have a limit at x = 2? Yes. See the analysis in a related problem.

Because the function is not defined at x = 2, it is not continuous. This discontinuity can be removed if the function is defined here by an additional statement: f(2) = 12. Then the value of the function will be the same value as its limit, satisfying the definition of continuity.