I need help figuring out if the function

f(x) = (x_1)^2 + e^(x_2) - e^(-x_2)

is coercive. I think not, but I’m not sure. I know that as x -> +/-infinity ||x|| -> infinity and lim f(x) = infinity. As x_2 -> infinity ||x|| -> infinity and lim f(x) = infinity. But, as x_2 -> -infinity ||x||-> infinity but lim f(x) = -infinity.

Does this definitively mean this function is not coercive?

Context: I’m trying to discover whether or not the exists a global min when the x \in R^2. If the function is coercive, that confirms that the global min does exist.
It is not coercive. Consider the parameterization γ(t)=(t, arcsinh(-t²/2)). Then |γ(t)| → ∞ but f(γ(t))=0 for all t.
My efforts are explained in the post.