Limit of multivariate polynomial with large arguments

Question

If I have a polynomial $f(x,y)=x^4+y^4-4xy$, how would I go about showing that as the standard norm of $(x,y)$ goes to infinity, $f(x,y)$ goes to infinity?

Answer 1

In the way that you've said it the statement is not correct. Consider $$ f(x, y) = x^2-y^2. $$ If you consider the line $x=y=t$, as $t\to\infty$, $x^2+y^2$ approaches $\infty$, but $f(x,y)$ is zero. So it's not true that as $\|(x,y)\|\to\infty$, $f(x,y)\to\infty$ as well.

I think you need to be more precise with the question statement; it's easy to modify it so that it becomes true.