For all $k \in \mathbb{N}$, we introduce the function $f_k : \mathbb{R}^n \rightarrow \mathbb{R}$ defined by
$$f_k(x) = f(x) + k\Psi(x) \text{ for all } x \in \mathbb{R}^n$$
where $\Psi : \mathbb{R}^n \rightarrow \mathbb{R}$ is the function defined by $\Psi(x) = \sum_{i=1}^{p} \max(0, g_i(x))^2$ for all $x \in \mathbb{R}^n$.\\
For all $x \in \mathbb{R}^n$, calculate $\lim_{k \rightarrow \infty} f_k(x)$.