Computing $\lim\limits_{(x,y)\to(0,0)} \frac{1-\cos(x^2+y^2)}{x^2+y^2}$ I need to calculate the following limit:
$$\lim\limits_{(x,y)\to(0,0)} \frac{1-\cos(x^2+y^2)}{x^2+y^2}$$
How does the limit go to 0? Please show steps.

Hint: $$\dfrac{1-\cos(x^2+y^2)}{x^2+y^2} = \dfrac{1-\cos^2(x^2+y^2)}{(x^2+y^2)\left(1+\cos(x^2+y^2)\right)} = \left(\dfrac{\sin(x^2+y^2)}{x^2+y^2}\right)^2\cdot\dfrac{x^2+y^2}{1+\cos(x^2+y^2)}.$$
