Algebra

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Find the horizontal and oblique asymptotes, if any, for the given rational function G(x)=(x^4-1)/(3x^2-3x)

  • Algebra -

    simplified ...
    f(x) = (x^2 + 1)(x-1)(x+1)/(3x(x-1))
    = (x+1)(x^2+1)/(3x)
    = (x^3 + x^2 + x + 1)/(3x)
    = (1/3) (x^2 + x + 1) + 1/(3x)

    clearly there is a vertical asymptote at x = 0
    there is also a "hole" at (1, 4/3) , (we divided top and bottom by x-1 )

    when x --> ∞
    1/3x --> 0
    so we left with
    (1/3)(x^2 + x + 1) + 0 , which is a paraloba

    so we have a "curved" asymptote formed by the parabola
    y = (1/3)(x^2 + x + 1)

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