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lim-x-x-2-x-2-4x-x-3-




Question Number 99951 by bemath last updated on 24/Jun/20
lim_(x→−∞)  x^2 (√(x^2 +4x)) + x^3  ?
$$\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\mathrm{x}^{\mathrm{2}} \sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{4x}}\:+\:\mathrm{x}^{\mathrm{3}} \:? \\ $$
Commented by bobhans last updated on 24/Jun/20
lim_(x→−∞)  (√(x^6 +4x^5 )) +x^3  ×{(((√(x^6 +4x^5 )) −x^3 )/( (√(x^6 +4x^5 ))−x^3 )) } =  lim_(x→−∞)  ((4x^5 )/( (√(x^6 +4x^5 ))−x^3 )) = lim_(x→−∞)  ((−4)/( (√((1/x^4 )+(4/x^5 )))+(1/x^2 ))) = −∞
$$\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\sqrt{\mathrm{x}^{\mathrm{6}} +\mathrm{4x}^{\mathrm{5}} }\:+\mathrm{x}^{\mathrm{3}} \:×\left\{\frac{\sqrt{\mathrm{x}^{\mathrm{6}} +\mathrm{4x}^{\mathrm{5}} }\:−\mathrm{x}^{\mathrm{3}} }{\:\sqrt{\mathrm{x}^{\mathrm{6}} +\mathrm{4x}^{\mathrm{5}} }−\mathrm{x}^{\mathrm{3}} }\:\right\}\:= \\ $$$$\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\frac{\mathrm{4x}^{\mathrm{5}} }{\:\sqrt{\mathrm{x}^{\mathrm{6}} +\mathrm{4x}^{\mathrm{5}} }−\mathrm{x}^{\mathrm{3}} }\:=\:\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\frac{−\mathrm{4}}{\:\sqrt{\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{4}} }+\frac{\mathrm{4}}{\mathrm{x}^{\mathrm{5}} }}+\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }}\:=\:−\infty \\ $$
Answered by mathmax by abdo last updated on 24/Jun/20
f(x) =x^2 (√(x^2 +4x)) +x^3  ⇒ for x<0  we get f(x) =x^2 ∣x∣(√(1+(4/x)))+x^3   =−x^3 (√(1+(4/x)))+x^3  ∼−x^3 (1+(2/x))+x^3  =−2x^2   (x→−∞) ⇒  lim_(x→−∞) f(x) =−∞
$$\mathrm{f}\left(\mathrm{x}\right)\:=\mathrm{x}^{\mathrm{2}} \sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{4x}}\:+\mathrm{x}^{\mathrm{3}} \:\Rightarrow\:\mathrm{for}\:\mathrm{x}<\mathrm{0}\:\:\mathrm{we}\:\mathrm{get}\:\mathrm{f}\left(\mathrm{x}\right)\:=\mathrm{x}^{\mathrm{2}} \mid\mathrm{x}\mid\sqrt{\mathrm{1}+\frac{\mathrm{4}}{\mathrm{x}}}+\mathrm{x}^{\mathrm{3}} \\ $$$$=−\mathrm{x}^{\mathrm{3}} \sqrt{\mathrm{1}+\frac{\mathrm{4}}{\mathrm{x}}}+\mathrm{x}^{\mathrm{3}} \:\sim−\mathrm{x}^{\mathrm{3}} \left(\mathrm{1}+\frac{\mathrm{2}}{\mathrm{x}}\right)+\mathrm{x}^{\mathrm{3}} \:=−\mathrm{2x}^{\mathrm{2}} \:\:\left(\mathrm{x}\rightarrow−\infty\right)\:\Rightarrow \\ $$$$\mathrm{lim}_{\mathrm{x}\rightarrow−\infty} \mathrm{f}\left(\mathrm{x}\right)\:=−\infty \\ $$

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