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lim-x-x-x-x-x-




Question Number 116007 by Khalmohmmad last updated on 30/Sep/20
lim_(x→∝) ((√x)/( (√(x+(√(x(√x)))))))
$$\underset{{x}\rightarrow\propto} {\mathrm{lim}}\frac{\sqrt{\mathrm{x}}}{\:\sqrt{\mathrm{x}+\sqrt{\mathrm{x}\sqrt{\mathrm{x}}}}} \\ $$
Answered by bemath last updated on 30/Sep/20
= (√(lim_(x→∞)  (x/(x+(√(x(√x))))))) =(√(lim_(x→∞)  (x/(x+(√(x^2 (((√x)/x))))))))   (√(lim_(x→∞)  (x/(x+x(√(√(1/x)))))))   = (1/(1+0)) = 1
$$=\:\sqrt{\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{{x}}{{x}+\sqrt{{x}\sqrt{{x}}}}}\:=\sqrt{\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{{x}}{{x}+\sqrt{{x}^{\mathrm{2}} \left(\frac{\sqrt{{x}}}{{x}}\right)}}}\: \\ $$$$\sqrt{\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{{x}}{{x}+{x}\sqrt{\sqrt{\frac{\mathrm{1}}{{x}}}}}}\: \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{1}+\mathrm{0}}\:=\:\mathrm{1} \\ $$$$ \\ $$
Answered by Dwaipayan Shikari last updated on 30/Sep/20
lim_(x→∞) ((√x)/( (√(x+(√(x(√x)))))))=(x^(1/2) /( x^(1/2) (√(1+x^(−(1/4)) ))))=(1/( (√(1+(1/x^(1/4) )))))=1
$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\sqrt{\mathrm{x}}}{\:\sqrt{\mathrm{x}+\sqrt{\mathrm{x}\sqrt{\mathrm{x}}}}}=\frac{\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} }{\:\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} \sqrt{\mathrm{1}+\mathrm{x}^{−\frac{\mathrm{1}}{\mathrm{4}}} }}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{4}}} }}}=\mathrm{1} \\ $$

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