Let $\displaystyle f(x,y)=sin|x|cos|y|$.

(a) Show from the limit definition that $\displaystyle f_y(0,0)$ exists, and find its value.

(b) Show from the limit definition that $\displaystyle f_y(0,0)$ does not exist.

My Attempt:

(a)

$\displaystyle \lim_{\Delta y \to 0} \frac{f(x_0, y_0 + \Delta y)-f(x_0,y_0)}{\Delta x}$

$\displaystyle \lim_{\Delta y \to 0} \frac{sin|0+|cos|0+\Delta x|-sin|0|xos|0|}{\Delta y}$

$\displaystyle \lim_{\Delta y \to 0} \frac{0}{\Delta y} = 0$

Is the value zero?

(b)

I can't prove that the limit doesn't exits because it turns out it exists:

$\displaystyle \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x, y_0)-f(x_0,y_0)}{\Delta x}$

$\displaystyle \lim_{\Delta x \to 0} \frac{sin|0+ \Delta x|cos|0|-sin|0|cos|0|}{\Delta x}$

$\displaystyle \lim_{\Delta x \to 0} \frac{sin|\Delta x|}{\Delta x} = 1$

(since limit as x ->0 sinx/x=1)

I'm confused can anyone help?