/sci/ - Science & Math

This function is continuous at the origin. I'm asked to determine whether or not it's also differentiable at the origin. I think it is, but I'm not 100% confident in my work:
[math]
\lim_{(x,y) \to (0,0)} \frac{\left| \frac{x^2y}{\sqrt{|xy|}}-0-\begin{pmatrix} 0 & 0 \end{pmatrix} \begin{pmatrix} x-0 \\ y-0 \end{pmatrix} \right|}{\| (x,y) \| }
= \lim_{(x,y) \to (0,0)} \frac{\left| \frac{x^2y}{\sqrt{|xy|}} \right|}{ \sqrt{x^2+y^2} }
= \lim_{(x,y) \to (0,0)} \frac{x^2\left|y\right|}{\sqrt{|xy|(x^2+y^2)}}
= \lim_{r \to 0^+} \frac{r^2\cos^2(\theta)\left|r\sin(\theta)\right|}{\sqrt{|r^2\cos(\theta)\sin(\theta)| \cdot r^2}}
= \lim_{r \to 0^+} \left|r\right| \frac{\cos ^2\left(\theta\right)\sqrt{\left|\sin \left(\theta\right)\right|}}{\sqrt{\left|\cos \left(\theta\right)\right|}}=0
[/math]
Thoughts?

This function is continuous at the origin. I'm asked to determine whether or not it's also differentiable at the origin. I think it is, but I'm not 100% confident in my work:
[math]
\lim_{(x,y) \to (0,0)} \frac{\left| \frac{x^2y}{\sqrt{|xy|}}-0-\begin{pmatrix} 0 & 0 \end{pmatrix} \begin{pmatrix} x-0 \\ y-0 \end{pmatrix} \right|}{\| (x,y) \| }
= \lim_{(x,y) \to (0,0)} \frac{\left| \frac{x^2y}{\sqrt{|xy|}} \right|}{ \sqrt{x^2+y^2} }
= \lim_{(x,y) \to (0,0)} \frac{x^2\left|y\right|}{\sqrt{|xy|(x^2+y^2)}}
= \lim_{r \to 0^+} \frac{r^2\cos^2(\theta)\left|r\sin(\theta)\right|}{\sqrt{|r^2\cos(\theta)\sin(\theta)| \cdot r^2}}
= \lim_{r \to 0^+} \left|r\right| \frac{\cos ^2\left(\theta\right)\sqrt{\left|\sin \left(\theta\right)\right|}}{\sqrt{\left|\cos \left(\theta\right)\right|}}=0.
[/math]

Thoughts?