I'm supposed to show whether the following limit exists or doesn't exist and if it exist calculate its value.
$$\lim_{(x,y,z) \to (0,0,0)} \frac{\ln(1+x^2+y^2+z^2)}{\sin(x^2+y^2+z^2)+xyz}$$
My approach has been to go along paths:
Along the path $x=y=z$ I get the limit to go to $1$ as follows :
$$\lim_{x \to 0} \frac{\ln(1+3x^2)}{\sin(3x^2)+x^3}$$
Using Taylor's approximation for $\ln$ and $\sin$, I simplified it to :
$$\lim_{x \to 0} \frac{3x^2}{3x^2+x^3}= 1 $$
Showing what the limit is along $1$ path is not enough to show that the limit exists. How should I continue (or start over with) this problem to officially show that the limit exists and goes to 1 or that it in fact does not?