Oblique asymptotes?A rational function, p ( x ) q ( x ) has an oblique asymptote only when the degree of p(x)= degree of q(x)−1.What "causes" the "slant" of the asymptote? Most asymptotes are caused by a function approaching an undefined value - I assume this is the same, but why (unlike others) would these asymptotes be slanted?Why does this only work with a difference of 1 between degrees?

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Oblique asymptotes?A rational function,             p      (      x      )              q      (      x      )       has an oblique asymptote only when the degree of p(x)= degree of q(x)−1.What &quot;causes&quot; the &quot;slant&quot; of the asymptote? Most asymptotes are caused by a function approaching an undefined value - I assume this is the same, but why (unlike others) would these asymptotes be slanted?Why does this only work with a difference of 1 between degrees?

vacchetta7k · Accepted Answer

It works with arbitrary degrees, it&#039;s just that with higher degrees the asymptotes won&#039;t be straight lines but parabolas etc.Consider an example:   f  (  x  )  =            4              x        3            +      1              2              x        2              =  2  x  +      1          2              x        2            , the latter part goes towards zero as x becomes large. You can do that with every such function - do polynomial long division and write out the residue. This will always look like   f  (  x  )  =            p      (      x      )              q      (      x      )        =  c  x  +  d  +            r      (      x      )              q      (      x      )       where c,d are constant and   d  e  g     r  &amp;lt;  d  e  g     q, so the latter fraction approaches zero as       |    x      |    →  ∞.

Tessa Peters · Accepted Answer

Write   p  (  x  )  =  (  a  +  b  x  )  ∗  q  (  x  )  +  r  (  x  ); dividing we have                    p        (        x        )                    q        (        x        )              =  a  +  b  x  +                    r        (        x        )                    q        (        x        )              ,where deg(r)&amp;lt;deg(q). The last term decays to zero as       |    x      |    →  ∞ and your oblique asymptote will be the line y=a+bx.You can have a difference larger than 1 of degrees. In this case, your graph will have the graph of a polynomial as an asymptote at   ±  ∞.

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