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lim-x-x-6-x-5-1-6-x-6-5x-5-1-6-

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Question Number 214258 by efronzo1 last updated on 03/Dec/24
lim_(x→∞) ((x^6 −x^5 ))^(1/6) −((x^6 +5x^5 ))^(1/6) =?
$$\:\:\:\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\sqrt[{\mathrm{6}}]{\mathrm{x}^{\mathrm{6}} −\mathrm{x}^{\mathrm{5}} }−\sqrt[{\mathrm{6}}]{\mathrm{x}^{\mathrm{6}} +\mathrm{5x}^{\mathrm{5}} }\:=? \\ $$
Answered by issac last updated on 03/Dec/24
−1
$$−\mathrm{1} \\ $$
Answered by golsendro last updated on 03/Dec/24
= lim_(x→0) ((((1−x))^(1/6) −((1+5x))^(1/6) )/x) = lim_(x→0) (((√(1−x))−(√(1+5x)))/(x(((1−x +(((1−x)(1+5x)))^(1/3) +((1+5x))^(1/3) ))^(1/3) ))) = lim_(x→0) (((√(1−x)) −(√(1+5x)))/(3x)) = lim_(x→0) ((−6x)/(3x((√(1−x))+(√(1+5x)) ))) = −1
$$\:\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\sqrt[{\mathrm{6}}]{\mathrm{1}−\mathrm{x}}\:−\sqrt[{\mathrm{6}}]{\mathrm{1}+\mathrm{5x}}}{\mathrm{x}}\: \\ $$$$\:\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}−\mathrm{x}}−\sqrt{\mathrm{1}+\mathrm{5x}}}{\mathrm{x}\left(\sqrt[{\mathrm{3}}]{\mathrm{1}−\mathrm{x}\:+\sqrt[{\mathrm{3}}]{\left(\mathrm{1}−\mathrm{x}\right)\left(\mathrm{1}+\mathrm{5x}\right)}\:+\sqrt[{\mathrm{3}}]{\mathrm{1}+\mathrm{5x}}}\right)} \\ $$$$\:\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}−\mathrm{x}}\:−\sqrt{\mathrm{1}+\mathrm{5x}}}{\mathrm{3x}} \\ $$$$\:\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{−\mathrm{6x}}{\mathrm{3x}\left(\sqrt{\mathrm{1}−\mathrm{x}}+\sqrt{\mathrm{1}+\mathrm{5x}}\:\right)} \\ $$$$\:\:=\:−\mathrm{1}\: \\ $$
Answered by golsendro last updated on 03/Dec/24
= lim_(x→0) ((((1−x))^(1/6) −((1+5x))^(1/6) )/x) = lim_(x→0) (((√(1−x))−(√(1+5x)))/(x(((1−x +(((1−x)(1+5x)))^(1/3) +((1+5x))^(1/3) ))^(1/3) ))) = lim_(x→0) (((√(1−x)) −(√(1+5x)))/(3x)) = lim_(x→0) ((−6x)/(3x((√(1−x))+(√(1+5x)) ))) = −1
$$\:\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\sqrt[{\mathrm{6}}]{\mathrm{1}−\mathrm{x}}\:−\sqrt[{\mathrm{6}}]{\mathrm{1}+\mathrm{5x}}}{\mathrm{x}}\: \\ $$$$\:\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}−\mathrm{x}}−\sqrt{\mathrm{1}+\mathrm{5x}}}{\mathrm{x}\left(\sqrt[{\mathrm{3}}]{\mathrm{1}−\mathrm{x}\:+\sqrt[{\mathrm{3}}]{\left(\mathrm{1}−\mathrm{x}\right)\left(\mathrm{1}+\mathrm{5x}\right)}\:+\sqrt[{\mathrm{3}}]{\mathrm{1}+\mathrm{5x}}}\right)} \\ $$$$\:\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}−\mathrm{x}}\:−\sqrt{\mathrm{1}+\mathrm{5x}}}{\mathrm{3x}} \\ $$$$\:\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{−\mathrm{6x}}{\mathrm{3x}\left(\sqrt{\mathrm{1}−\mathrm{x}}+\sqrt{\mathrm{1}+\mathrm{5x}}\:\right)} \\ $$$$\:\:=\:−\mathrm{1}\: \\ $$

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