2022 DHS Promo Q2

Given that \(\mathrm{f}\) is a continuous and increasing function, explain, with an aid of a sketch, why the value of \({\displaystyle \lim_{n\to\infty} \frac{2}{n}\left[\mathrm{f}(1)+\mathrm{f}\left(1+\frac{2}{n}\right)+\mathrm{f}\left(1+\frac{4}{n}\right)+\ldots+\mathrm{f}\left(1+\frac{2(n-1)}{n}\right)\right]}\) is \({\displaystyle \int_1^3 \mathrm{f}(x)\mathrm{d}x}\).[2]

Hence, evaluate \(\displaystyle \lim_{n\to\infty} \,\frac{2}{n}\left( {{\mathrm{e}}^{2}}+{{\mathrm{e}}^{2+\frac{4}{n}}}+{{\mathrm{e}}^{2+\frac{8}{n}}}+...+{{\mathrm{e}}^{2+\frac{4\left( n-1 \right)}{n}}} \right)\), leaving your answer in exact form.[2]