Use polar coordinates to find the indicated limit, if it exists. Note that is equivalent to .
A) 0
B) 1
C) -1
D) The limit is undefined.

Question

Use polar coordinates to find the indicated limit, if it exists. Note that  is equivalent to  . 
A) 0 
B) 1 
C) -1 
D) The limit is undefined.

Quizplus · Accepted Answer

Converting to polar coordinates, we have:
$$\lim_{(x,y)	o(0,0)} \frac{xy}{x^2 + y^2} = \lim_{r	o 0} \frac{r^2 \cos	heta \sin	heta}{r^2} = \lim_{r	o 0} \cos	heta \sin	heta$$
Note that this limit depends on the choice of $	heta$. However, since the limit is not the same for all choices of $	heta$, it does not exist. Therefore, the answer is D) The limit is undefined.

Use Polar Coordinates to Find the Indicated Limit, If It