Math Calc

Find the limit. (If the limit is infinite, enter '∞' or '-∞', as appropriate. If the limit does not otherwise exist, enter DNE.)

lim t-inf (sqrt(t)+t^2)/(9t-t^2)

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  1. the limit is -1
    Apply l'Hôpital's Rule twice

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  2. Divide numerator and denominator by t^2

    (sqrt(t)/t^2+t^2/t^2)/(9t/t^2-t^2/t^2) = (1/t^(3/2)+1)/(9/t -1)

    lim 1/t^(3/2)=0
    t->∞

    lim 9/t=0
    t->∞

    also

    lim 1/t^(3/2)=0
    t-> -∞

    lim 9/t=0
    t-> -∞

    So

    lim (1/t^(3/2)+1)/(9/t -1) =
    t-> -∞
    (0+1)/(0-1) = 1/ - 1 = - 1

    lim (1/t^(3/2)+1)/(9/t -1) =
    t-> ∞
    (0+1)/(0-1) = 1/ - 1 = - 1

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