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lim-x-x-x-2-3x-2-2022-x-x-2-5-2022-x-2022-




Question Number 162182 by bobhans last updated on 27/Dec/21
   lim_(x→∞)  (((x−(√(x^2 −3x+2)) )^(2022) +(x−(√(x^2 −5)) )^(2022) )/x^(2022) ) = ?
$$\:\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left(\mathrm{x}−\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{3x}+\mathrm{2}}\:\right)^{\mathrm{2022}} +\left(\mathrm{x}−\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{5}}\:\right)^{\mathrm{2022}} }{\mathrm{x}^{\mathrm{2022}} }\:=\:? \\ $$
Answered by cortano last updated on 27/Dec/21
  lim_(x→∞)  (((x−x(√(1−3x^(−1) +2x^(−2) )) )^(2022) +(x−x(√(1−5x^(−2) )) )^(2022) )/x^(2022) )    = lim_(x→∞)  (((1−(√(1−3x^(−1) +2x^(−2) )) )^(2022) +(1−(√(1−5x^(−2) )) )^(2022) )/1)    = 0
$$\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left({x}−{x}\sqrt{\mathrm{1}−\mathrm{3}{x}^{−\mathrm{1}} +\mathrm{2}{x}^{−\mathrm{2}} }\:\right)^{\mathrm{2022}} +\left({x}−{x}\sqrt{\mathrm{1}−\mathrm{5}{x}^{−\mathrm{2}} }\:\right)^{\mathrm{2022}} }{{x}^{\mathrm{2022}} } \\ $$$$\:\:=\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left(\mathrm{1}−\sqrt{\mathrm{1}−\mathrm{3}{x}^{−\mathrm{1}} +\mathrm{2}{x}^{−\mathrm{2}} }\:\right)^{\mathrm{2022}} +\left(\mathrm{1}−\sqrt{\mathrm{1}−\mathrm{5}{x}^{−\mathrm{2}} }\:\right)^{\mathrm{2022}} }{\mathrm{1}} \\ $$$$\:\:=\:\mathrm{0}\: \\ $$

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