高等数学（一）

<div> <h2>极限的四则运算法则</h2> <p>极限的四则运算法则是指在一定条件下，函数的极限运算可以按照加减乘除的顺序进行。</p> <ul> <li><strong>加法法则：</strong>若\(\lim_{{x \to a}} f(x) = A\)且\(\lim_{{x \to a}} g(x) = B\)，则\(\lim_{{x \to a}} [f(x) + g(x)] = A + B\)</li> <li><strong>减法法则：</strong>若\(\lim_{{x \to a}} f(x) = A\)且\(\lim_{{x \to a}} g(x) = B\)，则\(\lim_{{x \to a}} [f(x) - g(x)] = A - B\)</li> <li><strong>乘法法则：</strong>若\(\lim_{{x \to a}} f(x) = A\)且\(\lim_{{x \to a}} g(x) = B\)，则\(\lim_{{x \to a}} [f(x) \cdot g(x)] = A \cdot B\)</li> <li><strong>除法法则：</strong>若\(\lim_{{x \to a}} f(x) = A\)且\(\lim_{{x \to a}} g(x) = B\)，且\(B \neq 0\)，则\(\lim_{{x \to a}} \frac{f(x)}{g(x)} = \frac{A}{B}\)</li> </ul> <h3>例题说明</h3> <p>设\(f(x) = x^2\)，\(g(x) = 3x + 2\)，求\(\lim_{{x \to 2}} [f(x) + g(x)]\)</p> <ol> <li>首先计算\(\lim_{{x \to 2}} f(x)\)，即\(\lim_{{x \to 2}} x^2 = 4\)</li> <li>然后计算\(\lim_{{x \to 2}} g(x)\)，即\(\lim_{{x \to 2}} (3x + 2) = 8\)</li> <li>根据加法法则，\(\lim_{{x \to 2}} [f(x) + g(x)] = 4 + 8 = 12\)</li> </ol> </div>

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