If $f(x)=\bigg(1+\dfrac1x\bigg)^x,x>0$ then evaluate this limit:$$\lim_{n\to \infty}\bigg\{f(1/n)f(2/n)f(3/n)\dots f(n/n)\bigg\}^{1/n}$$
My attempt:
i rewrote this as $$y=\bigg\{\prod_{k=1}^n\bigg(1+\dfrac nk\bigg)^{k/n}\bigg\}^{1/n}$$
then i took $\ln$ on both sides but got stuck there!
Please tell how to proceed after this.
hint: Let $G(n)$ be the expression you want to evaluate the limit, then:
$\ln G(n) = \dfrac{1}{n}\left(\displaystyle \sum_{k=1}^n \ln f\left(\frac{k}{n}\right)\right)\implies \displaystyle \lim_{n\to \infty} \ln G(n) = \displaystyle \int_{0}^1 \ln f(x)dx$ .
Can you take it from here?