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#### LeibnizIsBetter

##### MHB Donor

- Aug 28, 2013

- 3

For example:

\(\displaystyle Prove \lim_{(x,y) \to (0,0)}\frac{2xy^2}{x^2+y^2} = 0\)

There are probably many ways to do this, but my teacher does it a certain way and I would like to learn his way first (although I am also interested in other techniques–perhaps using the polar coordinate system?).

He "massages" \(\displaystyle \left|f(x,y) - L\right| < \epsilon\) into a form compatible with \(\displaystyle \sqrt{(x-a)^2 + (y-b)^2} < \delta\) by using a series of inequalities.

\(\displaystyle

\left|\frac{2xy^2}{x^2 + y^2} - 0\right|

= \left|\frac{2\sqrt{x^2}y^2}{x^2 + y^2}\right|

\leq \left|\frac{2\sqrt{x^2 + y^2}y^2}{x^2 + y^2}\right|

= \left|\frac{2y^2}{\sqrt{x^2 + y^2}}\right|

\leq \left|\frac{2(y^2 + x^2)}{\sqrt{x^2 + y^2}}\right|

= 2\sqrt{x^2 + y^2}

\)

Step by step, I get this. But I'm not getting the bigger picture.

Could someone please explain to me how to use the above inequalities to prove the limit equals zero?

Thanks in advance. Any help is appreciated.