Answer
See explanation
Work Step by Step
The limit of a function $f$ is defined by: \[ \lim_{{(x, y, z) \to (x_0, y_0, z_0)}} f(x, y, z) = L \] If for any $\epsilon > 0$, we can find $\delta > 0$ such that \[ |\,f((x, y, z), (x_0, y_0, z_0)) - \delta| < \epsilon \quad |\,f(x, y, z) - L| < \epsilon \] $\epsilon$ represents the distance, which is strictly greater than the real distance between the actual value of the function and the value $L$. We can choose any $\epsilon > 0$, so even for very small values, this must hold. We need to find a value of $\delta > 0$ that satisfies \[ 0 < \sqrt{(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2} < \delta \] Here, $\delta$ represents the distance, which is strictly greater than the distance between the actual values of the coordinates $(x, y, z)$ and the coordinates of the limit $(x_0, y_0, z_0)$.
