Determine whether the sequence converges or diverges. If it converges, find the limit.
$$\left\{\frac{\ln4n}{\ln12n}\right\}$$
$\lim_{n\to \infty}\frac{\ln4n}{\ln12n} =$ ?
I am having trouble with this problem, any help is appreciated!!
[Math] Determine whether the sequence converges or diverges. If it converges, find the limit.
calculuslimitssequences-and-series
Best Answer
the limit is $1$. Write:$$\dfrac{\ln (4n)}{\ln (12n)} = \dfrac{\ln 4 + \ln n}{\ln 12 + \ln n} = \dfrac{1+\dfrac{\ln 4}{\ln n}}{1+\dfrac{\ln 12}{\ln n}}$$.
If instead you want to change the question to: $\displaystyle \lim_{n\to \infty} \dfrac{\ln(4+n)}{\ln(12+n)} = \text{ ?? }$, then you would apply the same trick as above:
$\ln(4+n) = \ln n + \ln\left(1+\dfrac{4}{n}\right)$, and divide top and bottom by $\ln n$ again as above.