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Math Help - More limits of natural logs.

  1. #1
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    More limits of natural logs.

    Why is the limit as x approaches +inf of (ln 2x)/(ln 3x) equal to 1? I thought that infinity/infinity was an indeterminate form? And I can't use L'Hopital's Rule yet either.


    Thank You.

    Jeannine
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  2. #2
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    Hello, Jeannine!

    Why does: . \lim_{x\to\infty} \frac{\ln(2x)}{\ln(3x)} \;=\;1 ?

    We have: . \lim_{x\to\infty} \frac{\ln(2) + \ln(x)}{\ln(3) + \ln(x)}


    Divide top and bottom by \ln(x)

    . . \lim_{x\to\infty}\frac{\frac{\ln(2)}{\ln(x)} + \frac{\ln(x)}{\ln(x)}} {\frac{\ln(3)}{\ln(x)} + \frac{\ln(x)}{\ln(x)}}\;= . \lim_{x\to\infty}\frac{\frac{\ln(2)}{\ln(x)} + 1}{\frac{\ln(3)}{\ln(x)} + 1} \;=\;\frac{0+1}{0+1} \;=\;1

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