Solving Limit of $\log {(n+1)}/\log {n}$

[steven187]

hello all

I was workin on a problem and during my proof I came to this problem does anybody know what happens with this limit

$$\lim_{n\rightarrow\infty} \frac{\log {(n+1)}}{\log {n}}$$

thanxs

 

[matt grime]

HINT: n+1 = n(1+1/n)

 

[steven187]

hello all

would this be correct?

$$\lim_{n\rightarrow\infty} \frac{\log {(n+1)}}{\log {n}}$$
$$=\lim_{n\rightarrow\infty} \frac{\log {n(1+\frac{1}{n})}}{\log {n}}$$
$$=\lim_{n\rightarrow\infty} \frac{\log{n}+\log {n(1+\frac{1}{n})}}{\log {n}}$$
$$=1+\lim_{n\rightarrow\infty} \frac{\log {(1+\frac{1}{n})}}{\log {n}}=1$$

 

[quetzalcoatl9]

yes that looks right except that you have a typo in the 3rd equation.