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
Hello, Jeannine!
Why does: .$\displaystyle \lim_{x\to\infty} \frac{\ln(2x)}{\ln(3x)} \;=\;1$ ?
We have: .$\displaystyle \lim_{x\to\infty} \frac{\ln(2) + \ln(x)}{\ln(3) + \ln(x)} $
Divide top and bottom by $\displaystyle \ln(x)$
. . $\displaystyle \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)}}\;= $ .$\displaystyle \lim_{x\to\infty}\frac{\frac{\ln(2)}{\ln(x)} + 1}{\frac{\ln(3)}{\ln(x)} + 1} \;=\;\frac{0+1}{0+1} \;=\;1$