I need to consulte one problem, just to control my result and see if I'm/ I'm not right:
I want to find
$$\frac{\partial f}{\partial x}(0,0), $$ where $f(x,y) = \mid xy \mid +\sin{xy}$ for $x,y \in \Re$
Because of the absolute value which hasn't got two-sided derivation in 0, I have to try to find it from definition of partial derivative in (0,0), which is defined as this limit
$$\lim_{t \to 0} \frac{\mid (t+x)0\mid + \sin{(x+t)0} -0}{t} = \lim_{t \to 0} \frac{0}{t} = 0$$.
So because the limit exists, the partial derivation exists and it is equal to 0.
Am I right, or is there some problem that I don't see?
Thank you very much!
You have a little mistake: you should write
$$\frac{\partial f}{\partial x}(0,0)=\lim_{t\to0}\frac{f(t,0)-f(0,0)}{t}=\lim_{t\to0}\frac{0}{t}=0$$