Maribel Ali

2023-03-06

How to find the Limit of $\mathrm{ln}\left(n+1\right)-\mathrm{ln}\left(n\right)$ as n approaches infinity?

cerisewi2

Use $\mathrm{ln}a-\mathrm{ln}b=\mathrm{ln}\left(\frac{a}{b}\right)$ and continuity of ln.
$\underset{n\to 00}{lim}\left(\mathrm{ln}\left(n+1\right)-\mathrm{ln}\left(n\right)=\underset{n\to \infty }{lim}\mathrm{ln}\left(\frac{n+1}{n}\right)$
$=\mathrm{ln}\left(\underset{n\to \infty }{lim}\frac{n+1}{n}\right)$
$=\mathrm{ln}\left(1\right)=0$
Note that
$\underset{n\to \infty }{lim}\frac{n+1}{n}=\underset{n\to \infty }{lim}\frac{\overline{)n}\left(1+\frac{1}{n}\right)}{\overline{)n}\cdot 1}$
$=\frac{1+0}{1}=1$

Do you have a similar question?