lim-x-1-x-1-p-1-x-1-q-1-

Question Number 121410 by john santu last updated on 07/Nov/20

\lim_{x \to 1} \frac{\sqrt[p]{x} - 1}{\sqrt[q]{x} - 1} ?

Answered by TANMAY PANACEA last updated on 07/Nov/20

x = t^{p q}

l i \underset{t \to 1}{m} \frac{t^{q} - 1}{t^{p} - 1}

l i \underset{t \to 1}{m} \frac{\frac{t^{q} - 1}{t - 1}}{\frac{t^{p} - 1}{t - 1}} = \frac{q \times {(1)}^{q - 1}}{p \times {(1)}^{p - 1}} = \frac{q}{p}

Answered by liberty last updated on 07/Nov/20

\lim_{x \to 1} \frac{x^{\frac{1}{p}} - 1}{x^{\frac{1}{q}} - 1} = \lim_{x \to 1} \frac{\frac{1}{p} x^{\frac{1}{p} - 1}}{\frac{1}{q} x^{\frac{1}{q} - 1}}

\lim_{x \to 1} \frac{{qx}^{\frac{1 - p}{p}}}{{px}^{\frac{1 - q}{q}}} = \frac{q}{p} .

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