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本帖最后由 hbghlyj 于 2022-9-24 13:16 编辑 artofproblemsolving.com/community/c7h490140p2748303
Calculate:
$\underset{x\to \infty }{\mathop{\lim }}\,\left( \sqrt[n]{{{P}_{n}}^{k}\left( x \right)}-\sqrt[n]{{{Q}_{n}}^{k}\left( x \right)} \right)$, where ${{P}_{n}}\left( x \right)={{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+...+{{a}_{1}}x+{{a}_{0}}\ ,\ {{Q}_{n}}\left( x \right)={{x}^{n}}+{{b}_{n-1}}{{x}^{n-1}}+...+{{b}_{1}}x+{{b}_{0}},{{P}_{n}}\left( x \right)\ne {{Q}_{n}}\left( x \right),$
$k=\min \left\{ p\in {{\mathbb{N}}^{*}},p\le n\left| {{a}_{n-p}}-{{b}_{n-p}}\ne 0 \right. \right\}.$
Proof:
$\underset{x\to \infty }{\mathop{\lim }}\,\left( \sqrt[n]{{{P}_{n}}^{k}\left( x \right)}-\sqrt[n]{{{Q}_{n}}^{k}\left( x \right)} \right)=\underset{x\to \infty }{\mathop{\lim }}\,\sqrt[n]{{{Q}_{n}}^{k}\left( x \right)}\left( \sqrt[n]{\frac{{{P}_{n}}^{k}\left( x \right)}{{{Q}_{n}}^{k}\left( x \right)}}-1 \right)=$
$\underset{x\to \infty }{\mathop{\lim }}\,\sqrt[n]{\frac{{{Q}_{n}}^{k}\left( x \right)}{{{x}^{nk}}}}\left( \frac{\sqrt[n]{\frac{{{P}_{n}}^{k}\left( x \right)}{{{Q}_{n}}^{k}\left( x \right)}}-1}{\frac{{{P}_{n}}\left( x \right)}{{{Q}_{n}}\left( x \right)}-1} \right){{x}^{k}}\left( \frac{{{P}_{n}}\left( x \right)}{{{Q}_{n}}\left( x \right)}-1 \right)=1\cdot \frac{k}{n}\cdot \left( {{a}_{n-k}}-{{b}_{n-k}} \right),$because
$\underset{x\to \infty }{\mathop{\lim }}\,\sqrt[n]{\frac{{{Q}_{n}}^{k}\left( x \right)}{{{x}^{nk}}}}=\underset{x\to \infty }{\mathop{\lim }}\,\sqrt[n]{{{\left( \frac{{{x}^{n}}+{{b}_{n-1}}{{x}^{n-1}}+...+{{b}_{1}}x+{{b}_{0}}}{{{x}^{n}}} \right)}^{k}}}=1,$
$\underset{x\to \infty }{\mathop{\lim }}\,\left( \frac{\sqrt[n]{\frac{{{P}_{n}}^{k}\left( x \right)}{{{Q}_{n}}^{k}\left( x \right)}}-1}{\frac{{{P}_{n}}\left( x \right)}{{{Q}_{n}}\left( x \right)}-1} \right)=\underset{x\to \infty }{\mathop{\lim }}\,\left( \frac{{{\left( \frac{{{P}_{n}}\left( x \right)}{{{Q}_{n}}\left( x \right)} \right)}^{\frac{k}{n}}}-1}{\frac{{{P}_{n}}\left( x \right)}{{{Q}_{n}}\left( x \right)}-1} \right)=\frac{k}{n},$
$\underset{x\to \infty }{\mathop{\lim }}\,{{x}^{k}}\left( \frac{{{P}_{n}}\left( x \right)}{{{Q}_{n}}\left( x \right)}-1 \right)=\underset{x\to \infty }{\mathop{\lim }}\,\frac{{{x}^{k}}\left( {{x}^{n-k}}\left( {{a}_{n-k}}-{{b}_{n-k}} \right)+...+\left( {{a}_{1}}-{{b}_{1}} \right)x+\left( {{a}_{0}}-{{b}_{0}} \right) \right)}{{{x}^{n}}+{{b}_{n-1}}{{x}^{n-1}}+...+{{b}_{1}}x+{{b}_{0}}}={{a}_{n-k}}-{{b}_{n-k}}.$ |
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