lim-x-a-x-1-m-a-1-m-x-1-n-a-1-n-Don-t-use-L-hospital-rules-

Question Number 29180 by A1B1C1D1 last updated on 05/Feb/18

\lim_{x \to a} (\frac{\sqrt[m]{x} - \sqrt[m]{a}}{\sqrt[n]{x} - \sqrt[n]{a}})

{Don}^{'} t use L^{'} hospital rules : -)

Answered by Rasheed.Sindhi last updated on 05/Feb/18

Formula

\lim_{x \to a} \frac{x^{n} - a^{n}}{x - a} = {na}^{n - 1}

\lim_{x \to a} (\frac{\sqrt[m]{x} - \sqrt[m]{a}}{\sqrt[n]{x} - \sqrt[n]{a}}) = \lim_{x \to a} \frac{x^{1 / m} - a^{1 / n}}{x^{1 / n} - a^{1 / n}}

= \lim_{x \to a} \frac{\frac{x^{1 / m} - a^{1 / n}}{x - a}}{\frac{x^{1 / n} - a^{1 / n}}{x - a}}

= \lim_{x \to a} \frac{\lim_{x \to a} \frac{x^{1 / m} - a^{1 / n}}{x - a}}{\lim_{x \to a} \frac{x^{1 / n} - a^{1 / n}}{x - a}} = \frac{\frac{1}{m} a^{\frac{1}{m} - 1}}{\frac{1}{n} a^{\frac{1}{n} - 1}}

= \frac{n}{m} a^{\frac{1}{m} - \frac{1}{n}} = \frac{n}{m} a^{\frac{n - m}{nm}}

Commented by A1B1C1D1 last updated on 05/Feb/18

Thanks