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Question Number 169527 by Giantyusuf last updated on 01/May/22

evaluate the limit(if it exists)  lim_(n→∞) [(((√(n^4 −2n^3 ))−n^2 )/(n+2))]^3

$$\boldsymbol{{evaluate}}\:\boldsymbol{{the}}\:\boldsymbol{{limit}}\left(\boldsymbol{{if}}\:\boldsymbol{{it}}\:\boldsymbol{{exists}}\right) \\ $$$$\boldsymbol{{lim}}_{\boldsymbol{{n}}\rightarrow\infty} \left[\frac{\sqrt{\boldsymbol{{n}}^{\mathrm{4}} −\mathrm{2}\boldsymbol{{n}}^{\mathrm{3}} }−\boldsymbol{{n}}^{\mathrm{2}} }{\boldsymbol{{n}}+\mathrm{2}}\right]^{\mathrm{3}} \\ $$

Commented by infinityaction last updated on 02/May/22

      p  =  lim_(n→∞)  [((n^2 ((√(1−(2/n))) −1))/(n+2))]^3        use binomail theorem for rational index       (1+x)^(n  ) = 1+nx+((n(n+1))/(2!))x^2 +.....  where −1 ⪇ x⪇1            p   =    lim_(n→∞)   [((1 −(1/n)  −1)/((1/n)(1+(2/n))))]^3             p    =    lim_(n→∞)  [ ((−1)/((1+(2/n))))]^3    =  −1

$$\:\:\:\:\:\:\boldsymbol{{p}}\:\:=\:\:\underset{{n}\rightarrow\infty} {\boldsymbol{\mathrm{lim}}}\:\left[\frac{\boldsymbol{{n}}^{\mathrm{2}} \left(\sqrt{\mathrm{1}−\frac{\mathrm{2}}{\boldsymbol{{n}}}}\:−\mathrm{1}\right)}{\boldsymbol{{n}}+\mathrm{2}}\right]^{\mathrm{3}} \\ $$$$\:\:\:\:\:\boldsymbol{\mathrm{use}}\:\boldsymbol{\mathrm{binomail}}\:\boldsymbol{\mathrm{theorem}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{rational}}\:\boldsymbol{\mathrm{index}} \\ $$$$\:\:\:\:\:\left(\mathrm{1}+\boldsymbol{\mathrm{x}}\right)^{\boldsymbol{\mathrm{n}}\:\:} =\:\mathrm{1}+\boldsymbol{\mathrm{nx}}+\frac{\boldsymbol{\mathrm{n}}\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)}{\mathrm{2}!}\boldsymbol{{x}}^{\mathrm{2}} +.....\:\:\boldsymbol{\mathrm{where}}\:−\mathrm{1}\:\lneq\:\boldsymbol{\mathrm{x}}\lneq\mathrm{1}\: \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{{p}}\:\:\:=\:\:\:\:\underset{{n}\rightarrow\infty} {\boldsymbol{\mathrm{lim}}}\:\:\left[\frac{\mathrm{1}\:−\frac{\mathrm{1}}{\boldsymbol{{n}}}\:\:−\mathrm{1}}{\frac{\mathrm{1}}{\boldsymbol{{n}}}\left(\mathrm{1}+\frac{\mathrm{2}}{\boldsymbol{{n}}}\right)}\right]^{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\boldsymbol{{p}}\:\:\:\:=\:\:\:\:\underset{{n}\rightarrow\infty} {\boldsymbol{\mathrm{lim}}}\:\left[\:\frac{−\mathrm{1}}{\left(\mathrm{1}+\frac{\mathrm{2}}{\boldsymbol{{n}}}\right)}\right]^{\mathrm{3}} \:\:\:=\:\:−\mathrm{1} \\ $$

Answered by cortano1 last updated on 02/May/22

 lim_(x→∞) ((((√(x^4 −2x^3 ))−x^2 )/(x+2)))^3 = (lim_(x→∞) ((x^2 (√(1−(2/x)))−x^2 )/(x+2)) )^3   = (lim_(x→∞) (((√(1−(2/x)))−1)/((1/x)+(2/x^2 ))) )^3    = (lim_(x→0)  (((√(1−2x))−1)/(x+2x^2 )))^3   =(lim_(x→0)  ((−2x)/(x(1+2x)((√(1−2x))+1))))^3   = (lim_(x→0)  ((−2)/((1+2x)((√(1−2x))+1))))^3   = (((−2)/(1(1+1))))^3 =−1

$$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\sqrt{{x}^{\mathrm{4}} −\mathrm{2}{x}^{\mathrm{3}} }−{x}^{\mathrm{2}} }{{x}+\mathrm{2}}\right)^{\mathrm{3}} =\:\left(\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{{x}^{\mathrm{2}} \sqrt{\mathrm{1}−\frac{\mathrm{2}}{{x}}}−{x}^{\mathrm{2}} }{{x}+\mathrm{2}}\:\right)^{\mathrm{3}} \\ $$$$=\:\left(\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\sqrt{\mathrm{1}−\frac{\mathrm{2}}{{x}}}−\mathrm{1}}{\frac{\mathrm{1}}{{x}}+\frac{\mathrm{2}}{{x}^{\mathrm{2}} }}\:\right)^{\mathrm{3}} \: \\ $$$$=\:\left(\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}−\mathrm{2}{x}}−\mathrm{1}}{{x}+\mathrm{2}{x}^{\mathrm{2}} }\right)^{\mathrm{3}} \\ $$$$=\left(\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{−\mathrm{2}{x}}{{x}\left(\mathrm{1}+\mathrm{2}{x}\right)\left(\sqrt{\mathrm{1}−\mathrm{2}{x}}+\mathrm{1}\right)}\right)^{\mathrm{3}} \\ $$$$=\:\left(\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{−\mathrm{2}}{\left(\mathrm{1}+\mathrm{2}{x}\right)\left(\sqrt{\mathrm{1}−\mathrm{2}{x}}+\mathrm{1}\right)}\right)^{\mathrm{3}} \\ $$$$=\:\left(\frac{−\mathrm{2}}{\mathrm{1}\left(\mathrm{1}+\mathrm{1}\right)}\right)^{\mathrm{3}} =−\mathrm{1}\: \\ $$

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