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lim-x-x-x-x-x-x-please-solution-




Question Number 186500 by 073 last updated on 05/Feb/23
lim_(x→+∞) (√(x+(√(x+(√(x+(√x)))))))−(√x)=?  please solution
$$\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}}}}}−\sqrt{\mathrm{x}}=? \\ $$$$\mathrm{please}\:\mathrm{solution} \\ $$
Answered by greougoury555 last updated on 05/Feb/23
 lim_(x→∞)  ((√(x+(√(x+(√x)))))/( (√(x+(√(x+(√(x+(√x))))))) +(√x)))   = lim_(x→∞)  (((√x) ((√(1+(√((1/x)+(√(1/x)))))) ))/( (√x) ((√(1+(√((1/x)+(√((1/x^2 )+(√(1/x^4 )))))))) +1)))  = (1/2)
$$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\sqrt{{x}+\sqrt{{x}+\sqrt{{x}}}}}{\:\sqrt{{x}+\sqrt{{x}+\sqrt{{x}+\sqrt{{x}}}}}\:+\sqrt{{x}}} \\ $$$$\:=\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\sqrt{{x}}\:\left(\sqrt{\mathrm{1}+\sqrt{\frac{\mathrm{1}}{{x}}+\sqrt{\frac{\mathrm{1}}{{x}}}}}\:\right)}{\:\sqrt{{x}}\:\left(\sqrt{\mathrm{1}+\sqrt{\frac{\mathrm{1}}{{x}}+\sqrt{\frac{\mathrm{1}}{{x}^{\mathrm{2}} }+\sqrt{\frac{\mathrm{1}}{{x}^{\mathrm{4}} }}}}}\:+\mathrm{1}\right)} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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