Got this limit I’d thought I should share it. So here you go:
$$\lim_{x\to\infty}{\bigg(\prod_{i=1}^{x}{\big(1+\frac{i}{x}\big)}\bigg)^{\frac{1}{x}}}$$
I have worked on this one for a while, with different angles and ended up with two different answers. Would like to see how you handled it.
On
$$y={\bigg(\prod_{i=1}^{x}{\big(1+\frac{i}{x}\big)}\bigg)^{\frac{1}{x}}}$$ $$y^x=\prod_{i=1}^{x}{\big(1+\frac{i}{x}\big)}$$ $$\log(y^x)=x \log(y)=\log{\bigg(\prod_{i=1}^{x}{\big(1+\frac{i}{x}\big)}\bigg)}=\sum_{i=1}^\infty\log\left(1+\frac i x\right)$$ $$\log(y)=\frac 1 x \sum_{i=1}^\infty\log\left(1+\frac i x\right)$$
Hint: Its logarithm is a Riemann sum.