Radially bounded or radially unbounded

Question

Show whether the following function is radially bounded or radially unbounded: $$ V(x)=\frac{(x_1+x_2)^2}{1+(x_1+x_2)^2}+(x_1-x_2)^2 $$

Answer 1

I this case, $V(x)$ is not radially unbounded (therefore radially bounded). Take each term individually.

Looking at the first term, $\frac{(x_1+x_2)^2}{1+(x_1+x_2)^2}$, this certainly does not go to infinity as the norm of $x$ goes to infinity. In fact, it actually goes to $1$.

Looking at the second term, $(x_1-x_2)^2$, it is not radially unbounded because the term does not go to infinity along the line $x_1 = x_2$. This wikipedia article covers this exact same case https://en.wikipedia.org/wiki/Radially_unbounded_function

Since neither term is radially unbounded, $V$ is radially bounded.