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find the verticle, horizontal, and oblique asymptotes, if any, for the given rational function: g(x)=x^4-81/3x^2-9x

I feel like such a dork becaue i have no clue how to do this, can someone please help me????

  • math -

    Vertical asymptote (v.a.) where the denominator is zero. That is, where 3x(x-3)=0. x=0 or 3

    But wait! If we factor top and bottom, we get
    g(x) = (x-3)(x+3)(x^2+9)/(3x)x-3))
    So, as long as x≠3,
    g(x) = (x+3)(x^2+9)/3x
    So, there's a hole at x=3 where g is undefined (because g(3) = 0/0), and the only v.a. is at x=0.

    For h.a., we need to see whether g(x) approaches some constant value as x gets huge. Now, for huge values of x, the only thing that matters is the highest power of x in the top and bottom. That means that for large x,

    g(x) =~ x^4/3x^2 = x^2/3

    So, this does not approach any constant value, nor does it approach any straight line. So, no h.a. or o.a.

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