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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?

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    $\begingroup$ Aren’t you really asking about rational functions, i.e. quotients of two polynomials? Seems to me that a polynomial function doesn’t have any asymptote at all, except in the trivial case of degree one. $\endgroup$ – Lubin Dec 3 '13 at 2:34
  • $\begingroup$ Yes, a rational function...sorry. $\endgroup$ – mepinon Dec 3 '13 at 2:39
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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.

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