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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
 Given a function f(x) = ((2ax^2 −x^3 ))^(1/3) .  Find the inclined asymptotes
$$\:{Given}\:{a}\:{function}\:{f}\left({x}\right)\:=\:\sqrt[{\mathrm{3}}]{\mathrm{2}{ax}^{\mathrm{2}} −{x}^{\mathrm{3}} }. \\ $$$${Find}\:{the}\:{inclined}\:{asymptotes} \\ $$
Answered by liberty last updated on 08/Dec/20
 k = lim_(x→±∞)  (y/x) = lim_(x→±∞) (((2ax^2 −x^3 ))^(1/3) /x)       lim_(x→±∞)  ((((2a)/x)−1))^(1/3)  = −1   b =lim_(x→±∞)  [y−kx ] = lim_(x→±∞) [ ((2ax^2 −x^3  ))^(1/3)  + x ]    = lim_(x→±∞)  ((2ax^2 −x^3 +x^3 )/( (((2ax^2 −x^3 )^2 ))^(1/3) −x ((2ax^2 −x^3 ))^(1/3)  +x^2 ))    = lim_(x→±∞) ((2ax^2 )/( (((2ax^2 −x^3 )^2 ))^(1/3) −x ((2ax^2 −x^3 ))^(1/3) +x^2 )) = ((2a)/3)  thus the straight line y=−x+((2a)/3) is a inclined  asymptotes to the curve y =((2ax^2 −x^3 ))^(1/3)  .
$$\:{k}\:=\:\underset{{x}\rightarrow\pm\infty} {\mathrm{lim}}\:\frac{{y}}{{x}}\:=\:\underset{{x}\rightarrow\pm\infty} {\mathrm{lim}}\frac{\sqrt[{\mathrm{3}}]{\mathrm{2}{ax}^{\mathrm{2}} −{x}^{\mathrm{3}} }}{{x}}\: \\ $$$$\:\:\:\:\underset{{x}\rightarrow\pm\infty} {\mathrm{lim}}\:\sqrt[{\mathrm{3}}]{\frac{\mathrm{2}{a}}{{x}}−\mathrm{1}}\:=\:−\mathrm{1} \\ $$$$\:{b}\:=\underset{{x}\rightarrow\pm\infty} {\mathrm{lim}}\:\left[{y}−{kx}\:\right]\:=\:\underset{{x}\rightarrow\pm\infty} {\mathrm{lim}}\left[\:\sqrt[{\mathrm{3}}]{\mathrm{2}{ax}^{\mathrm{2}} −{x}^{\mathrm{3}} \:}\:+\:{x}\:\right] \\ $$$$\:\:=\:\underset{{x}\rightarrow\pm\infty} {\mathrm{lim}}\:\frac{\mathrm{2}{ax}^{\mathrm{2}} −{x}^{\mathrm{3}} +{x}^{\mathrm{3}} }{\:\sqrt[{\mathrm{3}}]{\left(\mathrm{2}{ax}^{\mathrm{2}} −{x}^{\mathrm{3}} \right)^{\mathrm{2}} }−{x}\:\sqrt[{\mathrm{3}}]{\mathrm{2}{ax}^{\mathrm{2}} −{x}^{\mathrm{3}} }\:+{x}^{\mathrm{2}} } \\ $$$$\:\:=\:\underset{{x}\rightarrow\pm\infty} {\mathrm{lim}}\frac{\mathrm{2}{ax}^{\mathrm{2}} }{\:\sqrt[{\mathrm{3}}]{\left(\mathrm{2}{ax}^{\mathrm{2}} −{x}^{\mathrm{3}} \right)^{\mathrm{2}} }−{x}\:\sqrt[{\mathrm{3}}]{\mathrm{2}{ax}^{\mathrm{2}} −{x}^{\mathrm{3}} }+{x}^{\mathrm{2}} }\:=\:\frac{\mathrm{2}{a}}{\mathrm{3}} \\ $$$${thus}\:{the}\:{straight}\:{line}\:{y}=−{x}+\frac{\mathrm{2}{a}}{\mathrm{3}}\:{is}\:{a}\:{inclined} \\ $$$${asymptotes}\:{to}\:{the}\:{curve}\:{y}\:=\sqrt[{\mathrm{3}}]{\mathrm{2}{ax}^{\mathrm{2}} −{x}^{\mathrm{3}} }\:. \\ $$
Commented by bramlexs22 last updated on 08/Dec/20
thanks a lot
$${thanks}\:{a}\:{lot}\: \\ $$

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