問題：次の関数を微分せよ
- (1) \( \displaystyle \lim_{n \to \infty} \left(\dfrac{1}{n + 1} + \dfrac{1}{n + 2} + \cdots + \dfrac{1}{2n} \right) \)
- (2) \( \displaystyle \lim_{n \to \infty} \left(\dfrac{n}{n^2 + 1} + \dfrac{n}{n^2 + 4} + \cdots + \dfrac{1}{2n} \right) \)
- (3) \( \displaystyle \lim_{n \to \infty} \dfrac{1}{n} \left(\dfrac{1}{\sqrt{n^2 + 1}} + \dfrac{2}{\sqrt{n^2 + 4}} + \cdots + \dfrac{n - 1}{\sqrt{2n^2 - 2n + 1}} \right) \)
- (4) \( \displaystyle \lim_{n \to \infty} \dfrac{1}{n^2} \sum_{k=1}^{n} ke^{\frac{k}{n}} \)
- (5) \( \displaystyle \lim_{n \to \infty} \dfrac{\sqrt[n]{(n + 2)(n + 4) \cdots (3n)}}{n} \)