[Math] How many oblique asymptotes can a rational function have

Question

I know how to find the oblique/slant asymptote of a rational function. What I'm unsure about, because I've never dealt with this, is how many oblique asymptotes can a function actually have? And is it only possible to have an asymptote if the degree of the numerator polynomial and the degree of the denominator polynomial only differ by 1?

Answer 1

Those are actually called rational functions. An Oblique asymptote for one of those is the same at $\pm \infty.$

For other functions you can have two distinct oblique asymptotes, $$ \frac{\sqrt{1 + x^6}}{1 + x^2} $$ is roughly $|x|.$

Oh, my original point: you get at most two oblique asymptotes, because you are asking about the graph of a function.