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

Question

I am new to multivariabel limits and I have a seemingly difficult time trying to evaluate the limit in the question. I have tried changing to spherical coordinates and there after tried to show that the limit approaches infinity regardless of the angles $\theta$ and $\phi$.

My attempt:

$$\lim_{(x,y,z)\to (0,0,0)} \frac{x+y+z}{x^2+y^2+z^2}=\dots=\lim_{r \to 0^+}\frac{\sin(\theta)\cos(\phi)+\sin(\theta)\sin(\phi)+\cos(\theta)}{r}$$

Now this is where I have a hard time continuing since I don't know how to conclude anything from this. (I know that the limit shouldn't exist from the answers).

Answer 1

Try

1) $x=y=z=t$ , $t \rightarrow 0^+$ .

2) $x=y=z=t$ , $t \rightarrow 0^-$.

What happens?

Answer 2

Let $x=y=z$ and consider $x>0$, then the limit reduces to (if it exists) $\lim_{x\rightarrow 0^{+}}\dfrac{3x}{3x^{2}}=\infty$, so it does not exist.

Answer 3

Actually, your attempt will work, although there are easier approaches, as shown in other answers. If the limit exists, it must be the same no matter what $\theta$ and $\phi$ are. So you take $\theta=\phi=0,$ the fraction becomes $1/r,$ and the limit clearly doesn't exist.