<ol class="i-list"><li><div class="list-item-paragraph">Given that $\mathrm{f}$ is a continuous function, explain, with the aid of a sketch, why the value of</div><div class="list-item-paragraph" style="text-align: center">${\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}\left[ \mathrm{f}\left( 0 \right)+\mathrm{f}\left( \frac{1}{n} \right)+\mathrm{f}\left( \frac{2}{n} \right)+\ldots +\mathrm{f}\left( 1-\frac{1}{n} \right) \right]}$</div><div class="list-item-paragraph">is $\int_{0}^{1}{\mathrm{f}\left( x \right)}\ \mathrm{d}x$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Hence, evaluate ${\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{1}{n}\left( 1+{{\mathrm{e}}^{\frac{2}{n}}}+{{\mathrm{e}}^{\frac{4}{n}}}+\ldots +{{\mathrm{e}}^{\left( 2-\frac{2}{n} \right)}} \right) \right]}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">For any given positive integer $n$, explain whether ${\frac{1}{n}\left( 1+{{\mathrm{e}}^{\frac{2}{n}}}+{{\mathrm{e}}^{\frac{4}{n}}}+\ldots +{{\mathrm{e}}^{\left( 2-\frac{2}{n} \right)}} \right)}$ is larger or smaller than ${\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{1}{n}\left( 1+{{\mathrm{e}}^{\frac{2}{n}}}+{{\mathrm{e}}^{\frac{4}{n}}}+\ldots +{{\mathrm{e}}^{\left( 2-\frac{2}{n} \right)}} \right) \right]}$.</div><p style="text-align: right;">[1]</p></li></ol>

Given that $\mathrm{f}$ is a continuous function, explain, with the aid of a sketch, why the value of

${\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}\left[ \mathrm{f}\left( 0 \right)+\mathrm{f}\left( \frac{1}{n} \right)+\mathrm{f}\left( \frac{2}{n} \right)+\ldots +\mathrm{f}\left( 1-\frac{1}{n} \right) \right]}$

is $\displaystyle\displaystyle \int_{0}^{1}{\mathrm{f}\left( x \right)}\ \mathrm{d}x$.

Hence, evaluate ${\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{1}{n}\left( 1+{{\mathrm{e}}^{\frac{2}{n}}}+{{\mathrm{e}}^{\frac{4}{n}}}+\ldots +{{\mathrm{e}}^{\left( 2-\frac{2}{n} \right)}} \right) \right]}$.

For any given positive integer $n$, explain whether ${\frac{1}{n}\left( 1+{{\mathrm{e}}^{\frac{2}{n}}}+{{\mathrm{e}}^{\frac{4}{n}}}+\ldots +{{\mathrm{e}}^{\left( 2-\frac{2}{n} \right)}} \right)}$ is larger or smaller than ${\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{1}{n}\left( 1+{{\mathrm{e}}^{\frac{2}{n}}}+{{\mathrm{e}}^{\frac{4}{n}}}+\ldots +{{\mathrm{e}}^{\left( 2-\frac{2}{n} \right)}} \right) \right]}$.

2021 DHS JC2 MYE P1 Q2

2021 DHS JC2 MYE P1 Q2

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