## anonymous 5 years ago lim(x->infinity) sqrt(x-sqrt(x-sqrt(x)))-sqrt(x)

1. anonymous

$\lim(x->infinity) \sqrt{x-\sqrt{x-\sqrt{x}}}-\sqrt{x}$

2. 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$

3. 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.

4. anonymous

answer is $-\frac{1}{2}$

5. anonymous

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

6. anonymous

ah, yes