anonymous
  • anonymous
lim(x->infinity) sqrt(x-sqrt(x-sqrt(x)))-sqrt(x)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
\[\lim(x->infinity) \sqrt{x-\sqrt{x-\sqrt{x}}}-\sqrt{x}\]
anonymous
  • anonymous
Hello, I'm thinking you might want to try asymptotic equivalence to show first that\[x-\sqrt{x} \]~\[x\] (i.e. x-sqrt{x} is asymptotically equivalent to x). The two are as.eq. since\[\lim_{x-> \infty}\frac{x-\sqrt{x}}{x}=\lim_{x-> \infty}\frac{1-x^{-1/2}}{1}=1\]That means, as x approaches infinity, \[\sqrt{x-\sqrt{x-\sqrt{x}}} \iff \sqrt{x-\sqrt{x}}\iff \sqrt{x}\] which means,\[\lim_{n->\infty}\sqrt{x-\sqrt{x-\sqrt{x}}}-\lim_{n->\infty}\sqrt{x}\]\[\iff\lim_{n->\infty}\sqrt{x}-\lim_{n->\infty}\sqrt{x}=0\]
anonymous
  • anonymous
Where it should be noted that the logical equivalence symbol \[\iff\]should really be ~ since we're dealing with equivalence relations. I couldn't use the proper symbol in the equation editor for some reason.

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nikvist
  • nikvist
answer is \[-\frac{1}{2}\]
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anonymous
  • anonymous
thanks for the response.. i'm understand now.. :)
anonymous
  • anonymous
ah, yes

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