limits - Evaluate $limlimits_{n to infty} frac 1{nlog left ( 1+frac1n right ) }$

$$\lim\limits_{n \to \infty} \frac 1{n\log \left ( 1+\frac1n \right ) }$$

I start by writing $\lim\limits_{n \to \infty} \frac 1n=0$, and $\lim\limits_{n \to \infty} \frac 1{n+\log \left ( 1\frac1n \right ) }=0$

Since limit exists, by multiplication law, $\lim\limits_{n \to \infty} \frac 1{n\log \left ( 1+\frac1n \right ) }=0$

Hence the series $\sum_{n}^{\infty }\frac{1}{n\log 1+\frac{1}{n}}$ converge.

However, the $\lim n \to \infty \frac{1}{n\log \left ( 1+\frac{1}{n} \right ) }$ is given to be 1 in the solution, would really like to know where I've done wrong.

Thanks for the help.

Answer

Your first limit $$\lim_{n\to\infty}\frac{1}{n+\log(1+\frac{1}{n})}=0$$
In the corrected case we have $$\lim_{n\to \infty}\frac{1}{\log\left(1+\frac{1}{n}\right)^n}=1$$ since $$\lim_{n\to \infty}\log\left(1+\frac{1}{n}\right)^n=\log(e)=1$$ if $\log$ means logarithmus naturalis, to the base $e$