Tìm giới hạn \(A=\lim _{x \rightarrow+\infty} \frac{a_{0} x^{n}+\ldots+a_{n-1} x+a_{n}}{b_{0} x^{m}+\ldots+b_{m-1} x+b_{m}},\left(a_{0}, b_{0} \neq 0\right) .\)
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Lời giải:
Báo saiTa có:
\(A=\lim _{x \rightarrow+\infty} \frac{x^{n}\left(a_{0}+\frac{a_{1}}{x}+\ldots+\frac{a_{n-1}}{x^{n-1}}+\frac{a_{n}}{x^{n}}\right)}{x^{m}\left(b_{0}+\frac{b_{1}}{x}+\ldots+\frac{b_{m-1}}{x^{m-1}}+\frac{b_{m}}{x^{m}}\right)}\)
\(\text { Nếu } m=n \Rightarrow A=\lim _{x \rightarrow+\infty} \frac{a_{0}+\frac{a_{1}}{x}+\ldots+\frac{a_{n-1}}{x^{n-1}}+\frac{a_{n}}{x^{n}}}{b_{0}+\frac{b_{1}}{x}+\ldots+\frac{b_{m-1}}{x^{m-1}}+\frac{b_{m}}{x^{m}}}=\frac{a_{0}}{b_{0}} \text { . }\)
\(\begin{aligned} &\text { Nếu } m>n \Rightarrow A=\lim _{x \rightarrow+\infty} \frac{a_{0}+\frac{a_{1}}{x}+\ldots+\frac{a_{n-1}}{x^{n-1}}+\frac{a_{n}}{x^{n}}}{x^{m-n}\left(b_{0}+\frac{b_{1}}{x}+\ldots+\frac{b_{m-1}}{x^{m-1}}+\frac{b_{m}}{x^{m}}\right)}=0\\ &\text { ( Vì tử } \rightarrow a_{0}, \text { mẫu } \left.\rightarrow 0\right) \end{aligned}\)
\(\text { Nếu } m<n, \text { ta có: } A=\lim _{x \rightarrow+\infty} \frac{x^{n-m}\left(a_{0}+\frac{a_{1}}{x}+\ldots+\frac{a_{n-1}}{x^{n-1}}+\frac{a_{n}}{x^{n}}\right)}{b_{0}+\frac{b_{1}}{x}+\ldots+\frac{b_{m-1}}{x^{m-1}}+\frac{b_{m}}{x^{m}}}=\left\{\begin{array}{l} +\infty \text { khi } a_{0} \cdot b_{0}>0 \\ -\infty \text { khi } a_{0} b_{0}<0 \end{array}\right. \text { . }\)
