无穷小量与无穷大量
无穷小量的性质
重要程度：7 分
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<h2>无穷小量的性质</h2>
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<strong>性质1：有限个无穷小量的代数和仍是无穷小量。</strong>
<p>例如，若$\lim_{x \to a} f(x) = 0$且$\lim_{x \to a} g(x) = 0$，则$\lim_{x \to a} [f(x) + g(x)] = 0$。</p>
<p><em>例题：</em>设$f(x) = x$，$g(x) = x^2$，求$\lim_{x \to 0} [f(x) + g(x)]$。</p>
<p>解：因为$\lim_{x \to 0} x = 0$且$\lim_{x \to 0} x^2 = 0$，所以$\lim_{x \to 0} (x + x^2) = 0$。</p>
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<strong>性质2：有界函数与无穷小量的乘积仍是无穷小量。</strong>
<p>设$f(x)$在$x=a$处是有界的，即存在常数$M > 0$，使得$|f(x)| \leq M$，且$\lim_{x \to a} g(x) = 0$，则$\lim_{x \to a} [f(x)g(x)] = 0$。</p>
<p><em>例题：</em>设$f(x) = \sin(x)$，$g(x) = x$，求$\lim_{x \to 0} f(x)g(x)$。</p>
<p>解：因为$\sin(x)$在$x=0$处是有界的（$|\sin(x)| \leq 1$），且$\lim_{x \to 0} x = 0$，所以$\lim_{x \to 0} (\sin(x) \cdot x) = 0$。</p>
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<strong>性质3：常数与无穷小量的乘积仍是无穷小量。</strong>
<p>设$c$是常数，且$\lim_{x \to a} f(x) = 0$，则$\lim_{x \to a} [c \cdot f(x)] = 0$。</p>
<p><em>例题：</em>设$f(x) = x$，求$\lim_{x \to 0} 3 \cdot f(x)$。</p>
<p>解：因为$\lim_{x \to 0} x = 0$，所以$\lim_{x \to 0} (3 \cdot x) = 0$。</p>
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<strong>性质4：有限个无穷小量的乘积仍是无穷小量。</strong>
<p>设$\lim_{x \to a} f_1(x) = 0, \lim_{x \to a} f_2(x) = 0, \ldots, \lim_{x \to a} f_n(x) = 0$，则$\lim_{x \to a} [f_1(x) \cdot f_2(x) \cdots f_n(x)] = 0$。</p>
<p><em>例题：</em>设$f(x) = x$，$g(x) = x^2$，求$\lim_{x \to 0} f(x) \cdot g(x)$。</p>
<p>解：因为$\lim_{x \to 0} x = 0$且$\lim_{x \to 0} x^2 = 0$，所以$\lim_{x \to 0} (x \cdot x^2) = 0$。</p>
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