How can I calculate this limit $\lim_{(x,y)\to(0,0)} \frac{xy(1-cos(x^2+y^2))}{(x^2+y^2)^{\frac{5}{2}}}$? How can I calculate this limit $$\lim_{(x,y)\to(0,0)} \dfrac{xy(1-cos(x^2+y^2))}{(x^2+y^2)^{\frac{5}{2}}}$$ at the origin? 
I tried to use the substitution $x ^2 + y^2=t$ but how can I evaluate the value of $xy$? I even tried to use polar coordinates but to avail.

Outline: Note that $(x-y)^2\ge 0$, so $|xy|\le \frac{1}{2}(x^2+y^2)$. One can also get this  from polar coordinates, for $|xy|=r^2|\cos\theta\sin\theta|=\frac{1}{2}r^2|\sin(2\theta)|\le \frac{r^2}{2}$.
Now you can comfortably let $t=x^2+y^2$. You will need to look at the behaviour of $1-\cos t$ near $0$. This can be taken care of mechanically by using L'Hospital's Rule. Or else one can use the beginning of the Maclaurin series for $\cos t$. Or else one can note that 
$1-\cos t=\frac{1-\cos^2 t}{1+\cos t}=\frac{\sin^2 t}{1+\cos t}$.
The conclusion will be that the limit is $0$. 
