lim-x-x-1-1-4-x-1-4-x-1-1-3-x-1-3-

Question Number 124802 by john_santu last updated on 06/Dec/20

\lim_{x \to \infty} \frac{\sqrt[4]{x + 1} - \sqrt[4]{x}}{\sqrt[3]{x + 1} - \sqrt[3]{x}} = ?

Answered by bramlexs22 last updated on 06/Dec/20

\lim_{x \to \infty} \frac{\sqrt[4]{x} (\sqrt[4]{1 + \frac{1}{x}} - 1)}{\sqrt[3]{x} (\sqrt[3]{1 + \frac{1}{x}} - 1)} =

\lim_{x \to \infty} \frac{1}{\sqrt[12]{x}} (\frac{\sqrt[4]{1 + \frac{1}{x}} - 1}{\sqrt[3]{1 + \frac{1}{x}} - 1}) = 0

Answered by Bird last updated on 06/Dec/20

f (x) = \frac{{(x + 1)}^{\frac{1}{4}} - x^{\frac{1}{4}}}{{(x + 1)}^{\frac{1}{3}} - x^{\frac{1}{3}}} \Rightarrow

f (x) = \frac{x^{\frac{1}{4}} {{(1 + \frac{1}{x})}^{\frac{1}{4}} - 1}}{x^{\frac{1}{3}} {{(1 + \frac{1}{x})}^{\frac{1}{3}} - 1}}

\sim \frac{1}{x^{\frac{1}{3} - \frac{1}{4}}} \times \frac{\frac{1}{4 x}}{\frac{1}{3 x}} (x \to + \infty)

f (x) \sim \frac{3}{4} \times \frac{1}{x^{\frac{1}{12}}} \to 0 (x \to + \infty)

\Rightarrow {l i m}_{x \to + \infty} f (x) = 0

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