Coerciveness of a function - help

Question

I'm trying to show that

$$f(x_1,x_2,x_3) = e^{x_1^2 + x_2^2} + (x_1^2 + x_2^2 + 3x_2)^{500}$$ is not coercive, but am struggling to see anything.

Any help is appreciated!

Answer 1

The trick lies in fixing $x_1,x_2$ whilst taking $\|x_3\| \to \infty$ for which $f \neq \infty$