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Math Help - Indeterminate Forms/L'Hospital's Rule Question (must be simple but I don't get it)

  1. #1
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    Indeterminate Forms/L'Hospital's Rule Question (must be simple but I don't get it)

    We're asked to find:

     \lim \limits_{t \to 0}\frac{5^t - 3^t}{t}

    The solution involves L'Hospital's Rule. In the solution manual, it displays the intermediate step as:

     \lim \limits_{t \to 0}\frac{5^t ln 5 - 3^t ln 3}{1}

    Which equals:

    ln 5 - ln 3 = ln 3/5

    Anyone have any clue how they (specifically) managed to get the  \frac{5^t ln 5 - 3^t ln 3}{1}? Must be an algebraic trick that isn't occurring to me...?

    Thanks in advance!
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  2. #2
    Super Member Random Variable's Avatar
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    That's just a direct use of L'Hospital's rule. The derivative of the numerator is  5^{t} \ln(5) - 3^{t} \ln(3) and the derivative of the denominator is 1.
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