Asymptotes of a rational functionWe have a function f ( x ) = 2 x 4 + 4 x 3 + 3 x 2 + 4 x − 4 x 3 − x 2 − 6 x How would I systematically go about finding the asymptotes of this function? I know how to find the asymptotes of for example log functions or functions with a square root in it, but I don't really know how to find them for this function.

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Asymptotes of a rational functionWe have a function  f  (  x  )  =                    2                  x          4                +        4                  x          3                +        3                  x          2                +        4        x        −        4                              x          3                −                  x          2                −        6        x            How would I systematically go about finding the asymptotes of this function? I know how to find the asymptotes of for example log functions or functions with a square root in it, but I don&#039;t really know how to find them for this function.

Jase Powell · Accepted Answer

The problem can be simplified by dividing the numerator and denominator by (x+2):                    2                  x          3                +        3        x        −        2                              x          2                −        3        x              Vertical asymptotes occur where y tends to infinity, which happens at the roots of the denominator:      x    2    −  3  x  =  x  (  x  −  3  )  =  0  There are two roots so the vertical asymptotes are x=0 and x=3.Horizontal asymptotes occur where x tends to infinity, which becomes clear by dividing the numerator and the denominator by the highest power of x:      lim          x      →      ∞                          2                  x          3                +        3        x        −        2                              x          2                −        3        x              =      lim          x      →      ∞                          2        +        3                  /                          x          2                −        2                  /                          x          3                            1                  /                x        −        3                  /                          x          2                      =  ∞  There is no limit so there are no horizontal asymptotes.Oblique asymptotes occur where the graph approaches the line (mx+b), which becomes clear by dividing the numerator and denominator:                    2                  x          3                +        3        x        −        2                              x          2                −        3        x              =  2  x  +  6   remains   21  x  −  2  There is a whole part so the oblique asymptote is y=2x+6.

We have a function f(x)=(2x4+4x3+3x2+4x−4)/(x3−x2−6x) How would I systematically go about finding the asymptotes of this function? I know how to find the asymptotes of for example log functions or functions with a square root in it, but I don't really know how to find them for this function.

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