<ol class="abc-list"><li><div class="list-item-paragraph">Given that&nbsp;\(\mathrm{f}\) is a continuous function, explain, with the aid of a sketch, why the value of\[\lim_{n \to \infty} \frac{2}{n} \left\{ \mathrm{f}\left(1 + \frac{2}{n}\right) + \mathrm{f}\left(1 + \frac{4}{n}\right) + \dots + \mathrm{f}\left(1 + \frac{2n}{n}\right) \right\}\]is \(\displaystyle \displaystyle\int\nolimits_1^3 \mathrm{f}(x) \, \mathrm{d}x\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence evaluate \(\lim\limits_{n \to \infty} \sum_{k=1}^n \ln\left(1 + \frac{2k}{n}\right) \frac{2}{n}\) exactly.<span class="inline-marks">[4]</span></div></li></ol>

Given that \(\mathrm{f}\) is a continuous function, explain, with the aid of a sketch, why the value of\n\n\[\lim_{n \to \infty} \frac{2}{n} \left\{ \mathrm{f}\left(1 + \frac{2}{n}\right) + \mathrm{f}\left(1 + \frac{4}{n}\right) + \dots + \mathrm{f}\left(1 + \frac{2n}{n}\right) \right\}\]\n\nis \(\displaystyle \displaystyle\int\nolimits_1^3 \mathrm{f}(x) \, \mathrm{d}x\).

Given that \(\mathrm{f}\) is a continuous function, explain, with the aid of a sketch, why the value of\[\lim_{n \to \infty} \frac{2}{n} \left\{ \mathrm{f}\left(1 + \frac{2}{n}\right) + \mathrm{f}\left(1 + \frac{4}{n}\right) + \dots + \mathrm{f}\left(1 + \frac{2n}{n}\right) \right\}\]is \(\displaystyle \displaystyle\int\nolimits_1^3 \mathrm{f}(x) \, \mathrm{d}x\).

Hence evaluate \(\lim_{n \to \infty} \sum_{k=1}^n \ln\left(1 + \frac{2k}{n}\right) \frac{2}{n}\) exactly.

Hence evaluate \(\lim\limits_{n \to \infty} \sum_{k=1}^n \ln\left(1 + \frac{2k}{n}\right) \frac{2}{n}\) exactly.

2025 NYJC P2 Q3

2025 NYJC P2 Q3

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