How do you solve $\lim_{n\to\infty}\left(\frac{\log(n+1)}{\log(n)}\right)^{n\log(n)}$

Question

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

As it’s an indetermination, I’ve tried to do

$\exp\left(\lim\limits_{n\to\infty}(n\log(n))\cdot(\log(n+1)/\log(n)-1)\right)$.

Limit is equal to $e$ but I don't know how to make the exponent limit $0$.

Answer 1

Hint: $$\left(\dfrac{\log(n+1)}{\log(n)}\right)^{n \log(n)}=\left(1+\dfrac{\log(n+1)-\log(n)}{\log(n)}\right)^{n \log(n)}=\left(1+\dfrac{\log(1+\dfrac{1}{n})}{\log(n)}\right)^{n \log(n)}=\left(1+\dfrac{\log(1+\dfrac{1}{n})}{\log(n)}\right)^{\dfrac{\log(n)}{\log(1+\dfrac{1}{n})} \times n \log(1+\frac{1}{n})}$$ and $$\lim_{n \to \infty} \dfrac{\log(1+\dfrac{1}{n})}{\log(n)}=0,\lim_{n \to \infty} n \log(1+\frac{1}{n})=1$$