Intuition for coercive functions

I have been working with $\Gamma$-convergence for some time now; it has lead me to wonder: What is the intuition behind coercive functions?

Coercive function, where I have met such things, is one that grows sufficiently fast as the absolute value of its argument grows.

For example: A function $f$ from a normed space $X$ to real numbers might be called coercive iff $\lim_{|x| \to \infty } f(x) = \infty$. This means that the function eventually grows to infinity as its argument grows.

Another definition I have seen is $$\lim_{|x| \to \infty } \frac{f(x)}{|x|} = \infty.$$ This means that the function grows to infinity faster than a linear function.

There are other definitions in the same spirit of the function growing to infinity "sufficiently fast", for some definition of "sufficiently fast".

I suggest checking the precise definition of a coercive function that you are using. The consider the simplest possible case where the condition is satisfied and the simplest case where it is broken; functions defined on the real line are often good candidates, but if your function is a bilinear form, consider functions defined on $\mathbb{R}^2$ instead.