# Find a not coercive function $f: \mathbb{R}^2 \to \mathbb{R}$ which satisfies: $\lim_{|x_1| \to \infty} {f(x_1, \alpha x_1)} = \infty$

I am self-studying "Introduction to nonlinear optimization" by Amir Beck, and after studying chapter two which is called "Optimality Conditions for Unconstrained Optimization" I came across the following problem:

Find a function $$f: \mathbb{R}^2 \to \mathbb{R}$$ which is not coercive and satisfies that for any $$\alpha \in \mathbb{R}$$: $$\lim_{|x_1| \to \infty} {f(x_1, \alpha x_1)} = \lim_{|x_2| \to \infty} {f(\alpha x_2, x_2)} = \infty$$

My thought process for solving this problem was:

The function has to be symmetric with respect to variables since when we replace $$x_1$$ with $$x_2$$ it acts the same.

Then I said to myself since the limit requires both variables to approach $$\infty$$ at the same time, maybe I should find a function that goes to $$-\infty$$ if one of the variables is constant and the other one approaches $$\infty$$. But then I realized this case is equivalent to setting $$\alpha = 0$$.

Then I became pretty sure that I should find a function such that when variables approach $$\infty$$ on a different trajectory than $$x_2 = \alpha x_1$$ for example on $$x_2 = x_1^2$$, it goes to $$-\infty$$. But I have no idea how to find such a function.

Is my thought process correct? If not, what is wrong with it? If it is correct, then what is the next step? I really appreciate it if you can help me.

The function has to go to infinity along any ray, but the rate is not specified. Neither do you have any continuity constraints. Thus, you may define a function for every half-ray through the origin (by which is meant a set of the form $$\{\lambda x : \|x\| = 1\}$$), but with slopes that, though always positive, become arbitrarily small.
as @Cloudscape guided me, I found the function: $$f(x_1, x_2) = \frac{\lvert x_1 \rvert * \lvert x_2 \rvert} {\lvert x_1 \rvert + \lvert x_2 \rvert}$$