Question

Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.) ln(5n) ln(15n) lim n→∞ ln(5n) ln(15n)

Answer 1

If the limand is $\ln(5n)\ln(15n)$, then the limit diverges. But if you mean $\dfrac{\ln(5n)}{\ln(15n)}$, we can do some manipulating to rewrite it as

$\displaystyle\lim_{n\to\infty}\frac{\ln(5n)}{\ln(15n)}=\lim_{n\to\infty}\frac{\ln5+\ln n}{\ln15+\ln n}=\lim_{n\to\infty}\frac{\frac{\ln5}{\ln n}+1}{\frac{\ln15}{\ln n}+1}$

$\ln n\to\infty$ as $n\to\infty$, so the fractional terms vanish and you're left with 1.