Let $f(x,y)=x^2-2xy+y^2$. I know this is not coercive as along the line $y=x$, when $||x|| ->\infty, f(x,x)=0$.
But I don't understand what is wrong with the following way.
$$f(x,y)=x^2-2xy+y^2=(x^2+y^2)\left(1-{2xy\over x^2+y^2}\right).$$
Then $\lim_{||x||\to \infty},{2xy\over x^2+y^2} $ approaches 0.
So
$\lim_{||x||\to \infty} f(x,y)=\lim_{||x||\to \infty}(x^2+y^2)(1-0)$ and hence approaches $+\infty$
Similarly for $f(x_1,x_2,x_3)=x_1^2+x_2^2+x_3^2-4x_14x_2$
I did as $\lim_{||x||\to \infty}(x_1^2+x_2^2+x_3^2)(1-{4x_14x_2 \over x_1^2+x_2^2+x_3^2})$ approaches to $\infty$
but even this is not coercive.
But this approach works to show $f(x,y)=x^4+y^4-3xy$ .
What am I doing wrong in the above two examples.
I am showing the functions are coercive when they are not