lim (x->1) $\frac{\sqrt[m]{x} – 1}{\sqrt[n]{x} – 1 }$

Question

lim (x->1) $\frac{\sqrt[m]{x} – 1}{\sqrt[n]{x} – 1 }$

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Valentina 7 ngày 2021-12-03T12:55:46+00:00 2 Answers 4 views 0

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    0
    2021-12-03T12:57:16+00:00

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    0
    2021-12-03T12:57:25+00:00

    Ta có

    $\underset{x \to 1}{\lim} \dfrac{\sqrt[m]{x} – 1}{\sqrt[n]{x} – 1} = \underset{x \to 1}{\lim} \dfrac{(x-1)(\sqrt[n]{x^{n-1}} + \sqrt[n]{x^{n-2}} + \cdots + \sqrt[n]{x} + 1)}{(x-1)(\sqrt[m]{x^{n-1}} + \sqrt[m]{x^{n-2}} + \cdots + \sqrt[m]{x} + 1)}$

    $= \underset{x \to 1}{\lim} \dfrac{\sqrt[n]{x^{n-1}} + \sqrt[n]{x^{n-2}} + \cdots + \sqrt[n]{x} + 1}{\sqrt[m]{x^{n-1}} + \sqrt[m]{x^{n-2}} + \cdots + \sqrt[m]{x} + 1}$

    $= \dfrac{n}{m}$

    Vậy 

    $\underset{x \to 1}{\lim} \dfrac{\sqrt[m]{x} – 1}{\sqrt[n]{x} – 1} = \dfrac{n}{m}$.

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