|
Is $$\lim_{n\rightarrow +\infty}\ln\left(\frac{n+1}{n}\right)=0?$$ Because it is $\ln(1+\frac{1}{n})$ and $\frac{1}{n}$ tends to $0$, since $n$ tends to infinity, so the limit becomes $\ln(1+0)=\ln(1)=0$. Is this right, or is there any remarkable limit related to this? P.S. I am not used to formatting yet, I didn't really understand the rules, but I did what I could. |
|||||||||
|
|
$\ln$ is continuous at $1$, $\ln 1 = 0$, and $\lim_n \frac{n+1}{n} = 1$, hence $\lim_n \ln(\frac{n+1}{n}) = 0$. |
|||
|
|
|
Yes, continuity of the logarithm lets you bring the limit inside the logarithm so $$\lim_{n\rightarrow \infty} \ln\left(\frac{n+1}{n}\right)=\ln\left(\lim_{n\rightarrow\infty}\left(\frac{n+1}{n}\right)\right)$$ and yes you can do this for any continuous function besides logarithm. |
|||
|
|
