Given a function f(x) = ((2ax^2 −x^3 ))^(1/3) . Find the inclined asymptotes


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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

	$t h a n k s a l o t$
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