Hey I need to show that the limit(x,y)->(0,0) of xy / sqrt(x^2 + y^2) = 0. I'm not sure how to start manipulating this as I haven't gotten anything useful yet. Some help to get me going would be nice. Thanks
I don't know about the epsilon delta stuff. But here is a "conventional" way to show that the limit exists, maybe you can build from this.
If you change it to polar coordinates, then the function becomes
The maximum possible value of sin and cos is 1, but as r --> 0, r causes the product to go to 0, so because the limit holds from all sides, it exists and equals 0.
To prove you need to show . So in your case, to prove you need to show .
This may prove difficult, so instead we will convert to polars. Note that , and . Then to prove this limit, we would have to show
.
Working on the second inequality we have
So if we let and reverse the process, you will have your proof.