# multivariable limits

• Feb 9th 2009, 05:51 PM
archidi
multivariable limits
Prove that:

....... $\lim_{ (x,y)\to (1,1)}{x^2y}=1$,by using the definition of the limit for a multivariable function
• Feb 9th 2009, 07:06 PM
NonCommAlg
Quote:

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 and thus $x^2 < 4.$

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