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? asked in Science & MathematicsMathematics · 8 years ago

Find the limit as r approaches 0?

((a^r + b^r)/2)^(1/r)

3 Answers

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  • 8 years ago
    Favorite Answer

    lim ((a^r + b^r)/2)^(1/r) (as r --> 0)

    LN lim = lim LN

    lim LN ((a^r + b^r)/2)^(1/r) =

    = lim (1/r)LN ((a^r + b^r)/2) (as r --> 0) is a indetermination form 0/0

    L'Hopital rule

    lim ((a^r LN(a) + b^r LN(b))/(a^r + b^r)) (as r --> 0)

    = LN(a)/2 + LN(b)/2 = LN √(ab)

    as

    LN lim ((a^r + b^r)/2)^(1/r) = LN √(ab)

    lim ((a^r + b^r)/2)^(1/r) = √(ab)

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  • Anonymous
    6 years ago

    43fdas

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  • 8 years ago

    L = lim ((a^r + b^r)/2)^(1/r)

    ......r-> 0

    lnL = lim ln(((a^r + b^r)/2)^(1/r))

    ........r-> 0

    lnL = lim (1/r)ln((a^r + b^r)/2)

    ........r-> 0

    lnL = lim ln((a^r + b^r)/2) / r...now we can use L'Hopital's rule

    ........r-> 0

    lnL = lim ((lna(a^r) + lnb(b^r))/2) / 1 = (lna + lnb)/2

    ........r-> 0

    e^(lnL) = e^((lna + lnb)/2)

    L = e^(lna/2)e^(lnb/2)

    L = (e^lna - e^(1/2))(e^lnb - e^(1/2))

    L = (a - e^(1/2))(b - e^(1/2))

    L = ab - (a - b)e^(1/2) + e

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