Calculate-lim-x-0-x-x-sin-x-tan-2-x-1-3-

Question Number 166801 by mathocean1 last updated on 28/Feb/22

C a l c u l a t e :

\underset{x \to 0}{l i m} \frac{x - \sqrt{x}}{\sqrt[3]{s i n (x) - {t a n}^{2} (x)}}

Commented by cortano1 last updated on 28/Feb/22

\lim_{x \to 0^{+}} \frac{x - \sqrt{x}}{\sqrt[3]{\sin x - \tan^{2} x}}

Commented by mathocean1 last updated on 28/Feb/22

y e a h

Answered by bobhans last updated on 28/Feb/22

\lim_{x \to 0^{+}} \frac{x - \sqrt{x}}{\sqrt[3]{x - x^{2}}} = - \lim_{x \to 0^{+}} \frac{x - \sqrt{x}}{\sqrt[3]{x^{2} - x}}

= - \lim_{x \to 0^{+}} \frac{x^{2} - x}{x^{2} - x} . \lim_{x \to 0^{+}} \frac{\sqrt[3]{{(x^{2} - x)}^{2}}}{x + \sqrt{x}}

= - \lim_{x \to 0^{+}} \frac{x \sqrt[3]{x} (\sqrt[3]{{(1 - x^{- 1})}^{2}})}{\sqrt{x} (\sqrt{x} + 1)}

= - \lim_{x \to 0^{+}} \frac{\sqrt[3]{x} \sqrt[3]{{(1 - x^{- 1})}^{2}}}{\sqrt{x} + 1} = 0

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