Solving limit with three variables

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Homework Statement



\lim_{(x,y,z) \to (0,0,0)} \frac {x^3 + y^3 + z^3} {x^2 + y^2 + z^2}

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The Attempt at a Solution



I believe the limit is going to 0, but I have yet to find a way to prove this is the case.
 
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Let x^2+y^2+z^2=r^2. Then as (x,y,z)->(0,0,0), r->0. But e.g. |x|<r. Do you see what I'm saying?
 
Dick said:
Let x^2+y^2+z^2=r^2. Then as (x,y,z)->(0,0,0), r->0. But e.g. |x|<r. Do you see what I'm saying?

Not quite. I understand what you are doing with the bottom, the equation is that of a sphere, and therefore you are using another equation to show the bottom is going to 0. But how exactly does that prove the numerator is going to 0 as well?
 
If |x|<r and |y|<r and |z|<r, what about |x^3+y^3+z^3|?
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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