高等数学（工本）

<div> <h2>高阶无穷小、低阶无穷小、同阶无穷小、等价无穷小的概念</h2> <p><strong>1. 高阶无穷小</strong></p> <p>若$\lim\limits_{x \to x_0} \frac{\alpha(x)}{\beta(x)} = 0$，则称$\alpha(x)$是比$\beta(x)$高阶的无穷小，记作$\alpha(x) = o(\beta(x))$。</p> <p>例如：当$x \to 0$时，$x^2$是比$x$高阶的无穷小，即$x^2 = o(x)$。</p> <p><strong>2. 低阶无穷小</strong></p> <p>若$\lim\limits_{x \to x_0} \frac{\alpha(x)}{\beta(x)} = \infty$，则称$\alpha(x)$是比$\beta(x)$低阶的无穷小。</p> <p>例如：当$x \to 0$时，$x$是比$x^2$低阶的无穷小。</p> <p><strong>3. 同阶无穷小</strong></p> <p>若$\lim\limits_{x \to x_0} \frac{\alpha(x)}{\beta(x)} = c \neq 0$，则称$\alpha(x)$与$\beta(x)$是同阶无穷小。</p> <p>例如：当$x \to 0$时，$2x$与$x$是同阶无穷小，因为$\lim\limits_{x \to 0} \frac{2x}{x} = 2$。</p> <p><strong>4. 等价无穷小</strong></p> <p>若$\lim\limits_{x \to x_0} \frac{\alpha(x)}{\beta(x)} = 1$，则称$\alpha(x)$与$\beta(x)$是等价无穷小，记作$\alpha(x) \sim \beta(x)$。</p> <p>例如：当$x \to 0$时，$\sin x$与$x$是等价无穷小，因为$\lim\limits_{x \to 0} \frac{\sin x}{x} = 1$。</p> </div>

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