This has to be a rational equation in order to have those characteristics...in other words a fraction. One such equation would look like this:
x^2 - 2x - 8/x^2 - x - 12. When you factor both the numerator and the denominator you get (x-4)(x+2)/(x-4)(x+3). Because the (x-4) is in both the top and the bottom of the fraction, you can cancel them out, which makes that (x-4) a removable discontinuity. The (x+3) exists as a vertical asymptote.
