Given-a-function-f-x-2ax-2-x-3-1-3-Find-the-inclined-asymptotes-

Question Number 125099 by bramlexs22 last updated on 08/Dec/20

G i v e n a f u n c t i o n f (x) = \sqrt[3]{2 {a x}^{2} - x^{3}} .

F i n d t h e i n c l i n e d a s y m p t o t e s

Answered by liberty last updated on 08/Dec/20

k = \lim_{x \to \pm \infty} \frac{y}{x} = \lim_{x \to \pm \infty} \frac{\sqrt[3]{2 {a x}^{2} - x^{3}}}{x}

\lim_{x \to \pm \infty} \sqrt[3]{\frac{2 a}{x} - 1} = - 1

b = \lim_{x \to \pm \infty} [y - k x] = \lim_{x \to \pm \infty} [\sqrt[3]{2 {a x}^{2} - x^{3}} + x]

= \lim_{x \to \pm \infty} \frac{2 {a x}^{2} - x^{3} + x^{3}}{\sqrt[3]{{(2 {a x}^{2} - x^{3})}^{2}} - x \sqrt[3]{2 {a x}^{2} - x^{3}} + x^{2}}

= \lim_{x \to \pm \infty} \frac{2 {a x}^{2}}{\sqrt[3]{{(2 {a x}^{2} - x^{3})}^{2}} - x \sqrt[3]{2 {a x}^{2} - x^{3}} + x^{2}} = \frac{2 a}{3}

t h u s t h e s t r a i g h t l i n e y = - x + \frac{2 a}{3} i s a i n c l i n e d

a s y m p t o t e s t o t h e c u r v e y = \sqrt[3]{2 {a x}^{2} - x^{3}} .

Commented by bramlexs22 last updated on 08/Dec/20

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