So, we factor out the highest powers of x from both top and bottom polynomials, cancel, and then apply limit laws again. Similar ideas apply to any other fractions which involve polynomials and possibly roots. If the infinite limit laws produce expressions involving , we know that this doesn't make sense, so we look for a different approach. If polynomials are involved, factoring out the highest power is a good start. If square roots are involved, rationalizing might help.
L is such an asymptote.
Be careful nearby the vertical asymptotes to reflect whether a given one-sided limit is or , correspondingly. It doesn't hurt to plot several other points in each interval between asymptotes and x -intercepts, to make your graph more precise and make sure you haven't done any silly calculation mistakes in the above steps. We pray that such a problems is not on the midterm. If this doesn't help, we follow the steps below. Instead, we want x to be large enough.