Question about limit definition of partial derivative

[PNutMargarine]

I've seen it written two different ways:

$$\frac{\partial f}{\partial x} = \lim\limits_{h \rightarrow 0} \frac{f(x + h, y) - f(x,y)}{h}$$

and

$$\frac{\partial f}{\partial x} = \lim\limits_{h \rightarrow 0} \frac{f(x_0 + h, y_0) - f(x_0,y_0)}{h}$$

where the latter evaluates the function at the respective point before plugging it into the definition of the limit. For example, the function $f(x,y) = \begin{cases} \frac{x^2 y^4}{x^4 + 6y^8}, & \text{if }(x,y) \neq (0,0) \\ 0, & \text{if }(x,y) = (0,0) \end{cases}$

I want to determine if $\frac{\partial f}{\partial x}$ is differentiable at $(0,0)$.

Using the second limit definition would make showing the existence of $\frac{\partial f}{\partial x}$ considerably easier, since $y_0$ makes the first term in the limit $0$, and $f(x_0,y_0)$ is defined to be $0$.

But using the first definition, we have to evaluate:

$$\frac{(x+h)^2 y^4}{(x+h)^4 + 6y^8} - \frac{x^2 y^4}{x^4 + 6y^8}$$

I'm hoping the "real" or at least usable definition is the second one, but which one is the one we're supposed to use in practice to be technically/mathematically correct?

 

[Simon Bridge]

OTOH: there may be cases where $f(x_0,y_0)$ does not exist, but the limit does.
But otherwise you are fine:
http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfDerivative.aspx
... see example #4.