Question

Multivariable Limit

Answer 1

How have you used the hints?

Answer 2

I know that the limit exists using polar coordinates, but I am curious with the hint

Answer 3

\(\dfrac{2x(x^{2}-y^{2})}{x^{2}+y^{2}} = \dfrac{2|x|(x^{2}-y^{2})}{x^{2}+y^{2}}\) for \(x \ge 0\)

\(\dfrac{2|x|(x^{2}-y^{2})}{x^{2}+y^{2}} \le \dfrac{2\sqrt{x^{2} + y^{2}}(x^{2}-y^{2})}{x^{2}+y^{2}}\) -- Hint #1

\(\dfrac{2\sqrt{x^{2} + y^{2}}(x^{2}-y^{2})}{x^{2}+y^{2}} \le \dfrac{2\sqrt{x^{2} + y^{2}}(x^{2}+y^{2})}{x^{2}+y^{2}} = 2\sqrt{x^{2} + y^{2}}\) away from the Origin

Are we getting anywhere or just playing with symbols?

Answer 4

How about hint #2?

Answer 5

It's kind of in there. We just didn't need the absolute values.

Answer 6

okay, then we can write\[\lim_{(x,y)\rightarrow (0,0)} \frac{2x(x^2 -y^2)}{x^2 +y^2} \leq \lim_{(x,y)\rightarrow (0,0)} 2\sqrt{x^2+y^2}=0\]it is like we get the upper bound for the limit

Answer 7

Squeeze Theorem? I think you have it.

Answer 8

wait, doesn't it need the lower bound?

Answer 9

Well, that's where hint #2 comes in. You work on that.

Answer 10

oh yea, I got it now, thanks a lot