One-sided limits and existence of a two-sided limit

For $x \to 4^-$, we use the rule $c(x) = \sqrt{4-x}$ since this applies for $x \le 4$. As $x$ approaches 4 from the left, $4-x$ approaches $0^+$, so $\sqrt{4-x} \to 0$. Thus $\lim_{x\to 4^-} c(x) = 0$.

For $x \to 4^+$, we use the rule $c(x) = x-4$ since this applies for $x > 4$. As $x$ approaches 4 from the right, $x - 4$ approaches $0$. Thus $\lim_{x\to 4^+} c(x) = 0$ as well. Since both one-sided limits equal 0, the two-sided limit exists and $\lim_{x\to 4} c(x) = 0$.