高等数学（一）

<div> <h2>数列极限的四则运算法则</h2> <p>设$\{a_n\}$和$\{b_n\}$是两个收敛数列，且$\lim_{n \to \infty} a_n = A$，$\lim_{n \to \infty} b_n = B$，则有：</p> <ul> <li><strong>加法法则：</strong> $\lim_{n \to \infty} (a_n + b_n) = A + B$</li> <li><strong>减法法则：</strong> $\lim_{n \to \infty} (a_n - b_n) = A - B$</li> <li><strong>乘法法则：</strong> $\lim_{n \to \infty} (a_n \cdot b_n) = A \cdot B$</li> <li><strong>除法法则：</strong> 若$B \neq 0$，则$\lim_{n \to \infty} \left(\frac{a_n}{b_n}\right) = \frac{A}{B}$</li> </ul> <h3>例题1：加法法则</h3> <p>已知$\lim_{n \to \infty} \frac{1}{n} = 0$，$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right) = 1$。求$\lim_{n \to \infty} \left(\frac{1}{n} + 1 + \frac{1}{n}\right)$。</p> <p>解：根据加法法则，$\lim_{n \to \infty} \left(\frac{1}{n} + 1 + \frac{1}{n}\right) = \lim_{n \to \infty} \frac{1}{n} + \lim_{n \to \infty} 1 + \lim_{n \to \infty} \frac{1}{n} = 0 + 1 + 0 = 1$。</p> <h3>例题2：乘法法则</h3> <p>已知$\lim_{n \to \infty} \frac{1}{n} = 0$，$\lim_{n \to \infty} n = \infty$。求$\lim_{n \to \infty} \left(\frac{1}{n} \cdot n\right)$。</p> <p>解：根据乘法法则，$\lim_{n \to \infty} \left(\frac{1}{n} \cdot n\right) = \lim_{n \to \infty} \frac{1}{n} \cdot \lim_{n \to \infty} n = 0 \cdot \infty$。但这里需要特别注意，$\frac{1}{n} \cdot n = 1$，所以$\lim_{n \to \infty} \left(\frac{1}{n} \cdot n\right) = 1$。</p> <h3>例题3：除法法则</h3> <p>已知$\lim_{n \to \infty} \frac{1}{n} = 0$，$\lim_{n \to \infty} n = \infty$。求$\lim_{n \to \infty} \left(\frac{\frac{1}{n}}{n}\right)$。</p> <p>解：根据除法法则，$\lim_{n \to \infty} \left(\frac{\frac{1}{n}}{n}\right) = \frac{\lim_{n \to \infty} \frac{1}{n}}{\lim_{n \to \infty} n} = \frac{0}{\infty} = 0$。</p> </div>

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