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Find the limits or show that they do not exist:

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

Can anyone lend me a hand and teach me how to do this. I am totally at loss at this. :(

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Presumably this is multivariable calculus and homework. :) –  Ted Shifrin Jun 10 '13 at 13:28
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Hint: When the degree of the numerator and degree of the denominator are equal, you should think about what happens along different lines through the origin. –  Ted Shifrin Jun 10 '13 at 13:28
    
yes, i need to practice limits of functions of several variables and i came accros this question. do you know how to do? –  Risa Jun 10 '13 at 13:29
    
When the degree of the numerator and degree of the denominator are equal, their limits is 1? –  Risa Jun 10 '13 at 13:30
    
Don't just guess. Try different lines. What happens along $x=y=z$? What happens along $x=z=0$, etc.? –  Ted Shifrin Jun 10 '13 at 13:35
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2 Answers 2

Hint Let $$f(x,y,z)=\frac{xy+xz}{x^2+y^2+z^2}$$

then calculate $$\lim_{x\to0}f(x,0,0)\quad\text{and}\quad\lim_{x\to0}f(x,x,0)$$ what do you can conclude?

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lim x→0 f(x,0,0) = 0 lim x→0 f(x,x,0) = 1/2 since the limits are not the same, so the limit does not exist? –  Risa Jun 10 '13 at 13:42
    
Yes it's correct. –  Sami Ben Romdhane Jun 10 '13 at 13:44
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is that how u apply to all limits of function of 3 variables to get their limits? –  Risa Jun 10 '13 at 13:45
    
We use this method to show that the function hasn't a limit, but in the case when the function has a limit we prove this not give a counterexample. –  Sami Ben Romdhane Jun 10 '13 at 13:48
    
Do you know any examples of limits of function as (x,y,z) to (0,0,0)? –  Risa Jun 10 '13 at 13:51
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Note that $(x,y,z)\to0$ means $r:=\sqrt{x^2+y^2+z^2}\to0$. Introducing (geographical) spherical coordinates $\phi$, $\theta$ we have $$x=r\cos\phi\cos\theta,\quad y=r\sin\phi\cos\theta,\quad z=r\sin\theta$$ and therefore $${xy+xz\over x^2+y^2+z^2}=\cos\phi\cos\theta(\sin\phi\cos\theta+\sin\theta)\ .$$ This shows that the given expression is totally independent of $r>0$ and only depends on $\phi$ and $\theta$, i.e., on the "shadow" of the point $(x,y,z)$ on the unit sphere $S^2$. Since for all $\epsilon >0$ the "shadows" of points $(x,y,z)$ with $r<\epsilon$ still fill the full sphere the requested limit cannot exist.

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