How can I calculate this kind of limit?

Question

How can I calculate this limit?

$$\lim _{(x,y) \to (0,0)} \frac{\vert{x\vert\vert{y}\vert}}{x^2 +y^2}$$

I don't have idea and I will be appreciate for your help.

score 6 · Accepted Answer · answered Oct 19 '19 at 08:20

6

Suppose $\;x=0\;$ , then the limit is clearly zero, but if $\;y=x\;$ then

$$\frac{|x||y|}{x^2+y^2}=\frac{x^2}{2x^2}=\frac12\xrightarrow[x\to0]{}\frac12$$

Thus, the limit doesn't exist.

answered Oct 19 '19 at 08:20

DonAntonio

score 1 · Answer 2 · answered Oct 19 '19 at 08:10

1

Go polar to obtain

$$\frac{\vert{x\vert\vert{y}\vert}}{x^2 +y^2}=\vert{\cos \theta\vert\vert{\sin \theta}\vert}$$

answered Oct 19 '19 at 08:10

user

2 Answers2