What happens when a partial derivative can't be evaluated at a given point because of an indeterminacy? Is there any way to get around it?

Question

For example, I have the following function

$$f(x,y,z) = \begin{cases} \sqrt{|xyz|} & \text{if $(x,y,z) \ne (0,0,0)$} \\ 0 & \text{if $(x,y,z)=(0,0,0)$} \end{cases}$$

whose partial derivative with respect to x, if I followed the differentiation rules correctly, is

$$\frac{\partial f}{\partial x}(x,y,z) = \frac{xy^{2}z^{2}}{2|xyz|^{3/2}}\ $$

Now, I'm asked to evaluate the partial derivative at the origin (0,0,0); but as you can probably see, it's undefined at that point!

So what is there to be done? It's not the first exercise that leads me to this situation. Many thanks!

Answer 1

I think the partial derivatives simply do not exist. For simplicity, we can consider the function $$f(x) = \sqrt{|x|}.$$ At $x = 0$, the tangent line to the graph of the equation $y = f(x)$ is a vertical line, i.e. slope = $+/-\infty$. In other words, $f'(0)$ does not exist.

Answer 2

In this case, there is not a problem of indeterminacy: The partial derivative "blows up", going to $+\infty$ on the right and $-\infty$ on the left of the origin.

At any rate, nobody every promised that all functions should be everywhere differentiable.

You want indeterminacy? Try $$ \frac{\partial}{\partial x} \left( x \sin \left(\frac1{x} \right) \right)$$