<s>[INST] Limit multivarible How do I solve this limit?
$$\lim_{x,y,z) \to (0,0,0)} \frac{\sin(x^2+y^2+z^2)}{x^2+y^2+z^2+xyz} $$
This is equal to $$\frac{\sin(x^2+y^2+z^2)}{x^2+y^2+z^2}\times\frac1{1+\frac{xyz}{x^2+y^2+z^2}}$$
The first one is a standard limit with value one, but I'n not sure about the other term.
 [/INST] $0 \leq | \frac  {xyz} {x^{2}+y^{2}+z^{2}}|\leq \frac  1 2 |z| \to 0$ since $|xy| \leq \frac 1  2{(x^{2}+y^{2})}$. Hence the given limit is $1$.
</s>