How can I evaluate this multivariate limit?

Question

How can I evaluate the following limit? $$\lim_{|(x,y,z)|\to+\infty}(x^4+y^2+z^2-x+3y-z).$$

Answer 1

HINT

By completing the squares and squeeze theorem, note that

$$x^4+y^2+z^2-x+3y-z\ge x^2+y^2+z^2-x+3y-z=$$ $$=\left( x-\frac12\right)^2+\left( y+\frac32\right)^2+\left( z-\frac12\right)^2-\frac{11}4\to +\infty$$

Answer 2

Let $x=tu, \ y=tv, \ z=tw\,$ where $\,t = \sqrt{x^2+y^2+z^2}\,$ and $\,u=x/t, \ v=y/t, \ w=z/t$. Assuming $t>1$, we have that $$x^4+y^2+z^2-x+3y-z=t^2(t^2u^4+v^2+w^2)+t(-u+3v-w)\geq t^2(u^4+v^2+w^2)-t(|u|+3|v|+|w|)\geq t^2(t^2u^4+v^2+w^2)-5t.$$ Now, the function defined on the unit ball $\,\mathbb{S^2}\,$ by $(u,v,w)\mapsto u^4+v^2+w^2$ has an absolute minimum $p$ (by Weierstrass's theorem - $\,\mathbb{S^2}\,$ is closed and bounded). Also, $p>0$ because $u^4+v^2+w^2\geq0$, and if $u^4+v^2+w^2=0=p$ then $u=v=w$, but $(0,0,0)\notin \mathbb{S^2}$. Therefore $$x^4+y^2+z^2-x+3y-z>pt^2-5t$$ which clearly approaches $+\infty$ uniformly for $(u,v,w)\in \mathbb{S^2}$.

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Some Notes:

I used the following results:

$t^2>1\implies t^2u^2\geq u^2$ and $t(-u+3v-w)\geq-t|-u+3v-w|\geq-t(|u|+3|v|+|w|)$, as well as $|u|,|v|,|w|\leq 1.$