multivariable limits

**archidi** · February 9th 2009, 04:51 PM

Prove that:

....... $\lim_{ (x,y)\to (1,1)}{x^2y}=1$ ,by using the definition of the limit for a multivariable function

**NonCommAlg** · February 9th 2009, 06:06 PM

Originally Posted by archidi

Prove that:

....... $\lim_{ (x,y)\to (1,1)}{x^2y}=1$ ,by using the definition of the limit for a multivariable function

this is a good question! so suppose $\epsilon > 0$ is given. let $\delta=\min \{1, \frac{\epsilon}{7} \}.$ now if $\sqrt{(x-1)^2+(y-1)^2} < \delta,$ then: $|x-1|< \delta, \ |y-1| < \delta.$ since $\delta<1,$ we'll have $0<x<2$ and thus $x^2 < 4.$

therefore: $|x^2y-1|=|x^2(y-1)+x^2-1|<x^2|y-1|+(x+1)|x-1|<7\delta \leq \epsilon.$ hence $|x^2y-1| < \epsilon$ and we're done!

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