Consider a function $f(x): A \subset \mathbb{R} \to \mathbb{R}$ and another one $g(x)=f(x)+c$ with $c \in \mathbb{R}$.

Let $x\_0$ be a limit point for $A$. Is it the following true?

$$\exists \,\,\,\,\lim\_{x \to x\_0} f(x) \iff \exists \,\,\,\,\lim\_{x \to x\_0}g(x)$$