I’m given the operate $f : mathbb R^2 to mathbb R$ such that

$$(x,y)mapsto f(x,y) = |y|e^y(1+x^2y), $$

and I’m requested what situation I ought to impose on a closed, unbounded cone $C$ with vertex within the origin such that

$$lim_(x,y) f(x,y) = 0. $$

To reply this downside I first restricted myself to rays $(alpha t,beta t)$ with $alpha = costheta$, $beta = sintheta$ and $theta in [0,2pi)$, $t geq 0$. By tackling the 2 circumstances $beta > 0$, $beta < 0$ one sees that, because of the exponential, $|f|$ decays to $0$ alongside all rays, regardless of $alpha$. When $beta = 0$ the operate is trivially zero, so the identical holds.

Nonetheless, we aren’t allowed to conclude that every one cones are legitimate, since usually the truth that a two-variable operate behaves a way alongside all rays doesn’t suggest that there is not a pesky sequence resulting in a special consequence. And certainly, when the ray could be very near the $x$-axis on both facet, the restriction of $|f|$ to mentioned ray has a most (as a consequence of Weierstrass’s theorem: $f$ is steady and goes to $0$ as ) that will increase with out certain as $theta to 0^+, pi, 2pi^-$, which signifies that if I choose the pesky sequence

$$(x_n,y_n) = (t_ncos(1/n), t_nsin(1/n)),$$

with $t_n$ being the purpose the place $|f|$ attains mentioned most, then $|f(x_n,y_n)| to infty$. So this recommend that we must always prescribe that $C$ to keep away from the $x$-axis completely.

I need to justify the truth that there may be an rising most extra concretely than simply with instinct. I’ve expanded to first order in $theta$, for instance within the case $theta to 0^+$: for fastened $t geq 0$,

$$start{break up}

f(tcostheta,tsintheta) &= t(sintheta)e^{-tsintheta} (1 + t^3cos^2 theta sintheta)

&approx theta t (1 – theta t) (1+theta t^3) approx theta t,

finish{break up}$$

which I reckon signifies that as $theta$ approaches $0^+$, the operate alongside the ray is an increasing number of much like a operate that explodes linearly with $t$. Nonetheless, I am undecided about this argument and I do not assume it helps me.

Attempting to explicitly maximize $|f|$ for a set ray is out of the query, seeing because it requires me to unravel a quartic equation in $t$.

After some tinkering I’ve observed that proscribing $f$ to the double hyperbola $(x, pm 1/x)$ results in a non-zero restrict:

$$left| fleft(x,pm frac 1 xright) proper| = frac 1 x e^{-1/x} (1 pm x) approx frac{xpm 1}{x} xrightarrow{xtopminfty} 1, $$

which I suppose goes to show my declare (utilizing $x_n = n$, $y_n = pm 1/n$).

Questions:

- Is learning the restriction of $|f|$ to rays helpful in any respect?
- What’s an environment friendly option to show that the max of $|f|$ on a ray will increase with out certain because the ray approaches the $x$-axis?
- Is it doable to reach on the double hyperbola by learning the rays?
- How do I do know issues come up solely across the $x$-axis?