高等数学（工本）

<div> <h2>无穷小的性质</h2> <ul> <li><strong>性质1：有限个无穷小的代数和仍是无穷小。</strong> <p>例如，设\( \alpha(x) \)和\( \beta(x) \)都是当\( x \to x_0 \)时的无穷小，则\( \alpha(x) + \beta(x) \)也是当\( x \to x_0 \)时的无穷小。</p> <p>证明：设\( \lim_{x \to x_0} \alpha(x) = 0 \)，\( \lim_{x \to x_0} \beta(x) = 0 \)，则根据极限的性质，有：</p> <p>\( \lim_{x \to x_0} (\alpha(x) + \beta(x)) = \lim_{x \to x_0} \alpha(x) + \lim_{x \to x_0} \beta(x) = 0 + 0 = 0 \)</p> </li> <li><strong>性质2：有限个无穷小的乘积仍是无穷小。</strong> <p>例如，设\( \alpha(x) \)和\( \beta(x) \)都是当\( x \to x_0 \)时的无穷小，则\( \alpha(x) \cdot \beta(x) \)也是当\( x \to x_0 \)时的无穷小。</p> <p>证明：设\( \lim_{x \to x_0} \alpha(x) = 0 \)，\( \lim_{x \to x_0} \beta(x) = 0 \)，则根据极限的性质，有：</p> <p>\( \lim_{x \to x_0} (\alpha(x) \cdot \beta(x)) = \lim_{x \to x_0} \alpha(x) \cdot \lim_{x \to x_0} \beta(x) = 0 \cdot 0 = 0 \)</p> </li> <li><strong>性质3：有界函数与无穷小的乘积仍是无穷小。</strong> <p>例如，设\( f(x) \)在\( x_0 \)附近有界，即存在常数\( M > 0 \)，使得\( |f(x)| \leq M \)对所有\( x \)成立，且\( \alpha(x) \)是当\( x \to x_0 \)时的无穷小，则\( f(x) \cdot \alpha(x) \)也是当\( x \to x_0 \)时的无穷小。</p> <p>证明：设\( \lim_{x \to x_0} \alpha(x) = 0 \)，则根据极限的性质，有：</p> <p>\( \lim_{x \to x_0} (f(x) \cdot \alpha(x)) = \lim_{x \to x_0} f(x) \cdot \lim_{x \to x_0} \alpha(x) = \lim_{x \to x_0} f(x) \cdot 0 = 0 \)</p> </li> </ul> </div>

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