More limits of natural logs.

**jbecker007** · September 30th 2008, 06:54 PM

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

**Soroban** · September 30th 2008, 09:15 PM

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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