第四节 极限运算法则
极限四则运算法则
重要程度：8 分
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<h2>极限四则运算法则</h2>
<p><strong>1. 极限的加法法则：</strong></p>
<p>若 \(\lim\limits_{x \to a} f(x) = A\) 且 \(\lim\limits_{x \to a} g(x) = B\) ，则</p>
<p>\[\lim\limits_{x \to a} [f(x) + g(x)] = A + B\]</p>
<p><strong>例题：</strong></p>
<p>求 \(\lim\limits_{x \to 2} (3x^2 + 2x - 1)\)</p>
<p>解：根据加法法则，分别求 \(3x^2\) 和 \(2x-1\) 的极限。</p>
<p>\(\lim\limits_{x \to 2} 3x^2 = 3 \times 4 = 12\)，\(\lim\limits_{x \to 2} 2x - 1 = 2 \times 2 - 1 = 3\)</p>
<p>因此 \(\lim\limits_{x \to 2} (3x^2 + 2x - 1) = 12 + 3 = 15\)</p>
<p><strong>2. 极限的减法法则：</strong></p>
<p>若 \(\lim\limits_{x \to a} f(x) = A\) 且 \(\lim\limits_{x \to a} g(x) = B\) ，则</p>
<p>\[\lim\limits_{x \to a} [f(x) - g(x)] = A - B\]</p>
<p><strong>例题：</strong></p>
<p>求 \(\lim\limits_{x \to 1} (x^2 - 4x + 3)\)</p>
<p>解：根据减法法则，分别求 \(x^2\) 和 \(4x - 3\) 的极限。</p>
<p>\(\lim\limits_{x \to 1} x^2 = 1\)，\(\lim\limits_{x \to 1} 4x - 3 = 4 \times 1 - 3 = 1\)</p>
<p>因此 \(\lim\limits_{x \to 1} (x^2 - 4x + 3) = 1 - 1 = 0\)</p>
<p><strong>3. 极限的乘法法则：</strong></p>
<p>若 \(\lim\limits_{x \to a} f(x) = A\) 且 \(\lim\limits_{x \to a} g(x) = B\) ，则</p>
<p>\[\lim\limits_{x \to a} [f(x) \cdot g(x)] = A \cdot B\]</p>
<p><strong>例题：</strong></p>
<p>求 \(\lim\limits_{x \to 3} (2x \cdot (x+1))\)</p>
<p>解：根据乘法法则，分别求 \(2x\) 和 \(x+1\) 的极限。</p>
<p>\(\lim\limits_{x \to 3} 2x = 2 \times 3 = 6\)，\(\lim\limits_{x \to 3} x+1 = 3+1 = 4\)</p>
<p>因此 \(\lim\limits_{x \to 3} (2x \cdot (x+1)) = 6 \cdot 4 = 24\)</p>
<p><strong>4. 极限的除法法则：</strong></p>
<p>若 \(\lim\limits_{x \to a} f(x) = A\) 且 \(\lim\limits_{x \to a} g(x) = B\) 且 \(B \neq 0\) ，则</p>
<p>\[\lim\limits_{x \to a} \frac{f(x)}{g(x)} = \frac{A}{B}\]</p>
<p><strong>例题：</strong></p>
<p>求 \(\lim\limits_{x \to 2} \frac{x^2 - 1}{x - 1}\)</p>
<p>解：根据除法法则，分别求 \(x^2 - 1\) 和 \(x - 1\) 的极限。</p>
<p>\(\lim\limits_{x \to 2} x^2 - 1 = 2^2 - 1 = 3\)，\(\lim\limits_{x \to 2} x - 1 = 2 - 1 = 1\)</p>
<p>因此 \(\lim\limits_{x \to 2} \frac{x^2 - 1}{x - 1} = \frac{3}{1} = 3\)</p>
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