Question

lim(x->infinity) sqrt(x-sqrt(x-sqrt(x)))-sqrt(x)

Answer 1

\[\lim(x->infinity) \sqrt{x-\sqrt{x-\sqrt{x}}}-\sqrt{x}\]

Answer 2

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,\[\lim_{n->\infty}\sqrt{x-\sqrt{x-\sqrt{x}}}-\lim_{n->\infty}\sqrt{x}\]\[\iff\lim_{n->\infty}\sqrt{x}-\lim_{n->\infty}\sqrt{x}=0\]

Answer 3

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.

Answer 4

answer is \[-\frac{1}{2}\]

Answer 5

thanks for the response.. i'm understand now.. :)

Answer 6

ah, yes