📝 题目
例 3 设 $a > 1,k > 0$ ,求证: $\displaystyle{\mathop{\lim }\limits_{{x \rightarrow + \infty }}\frac{{x}^{k}}{{a}^{x}} = 0}$ .
💡 答案与解析
证 不妨设 $x > 1$ ,注意到 $\displaystyle{\mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{{n}^{k}}{{a}^{n}} = 0}$ ,则有
$$ 0 \leq \frac{{x}^{k}}{{a}^{x}} \leq \frac{{\left( \left\lbrack x\right\rbrack + 1\right) }^{k}}{{a}^{\left\lbrack x\right\rbrack }} = a \cdot \frac{{\left( \left\lbrack x\right\rbrack + 1\right) }^{k}}{{a}^{\left\lbrack x\right\rbrack + 1}} \Rightarrow \mathop{\lim }\limits_{{x \rightarrow + \infty }}\frac{{x}^{k}}{{a}^{x}} = 0. $$
\subsubsection{二、用变量代换}