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Recently I have been trying to reason why certain rational functions such as (-2(2x^2+x-31))/((x-3)*(x+4)*(x+1)) have varying near horizontal asymptote slopes. I know that the direction of the horizontal asymptote can be varied by altering the parent function x+1/x, but several equations like the one above do not duplicate the single line or single slope that variations of that parent equation produce.

So I quess my question is, in rational functions what causes variation in the slope of horizontal asymptotes?

I have been able to glean that when the number of roots on top of the rational function is greater than or equal to the number on the bottom, the slope of the asymptotes is something besides 0. Also it seems that the slopes of the horizontal asymptotes vary from each other when there are 1 or more complex roots in the dividend of the equation.

Thanks,

-scott