Answer
$$\lim_{x \to 5} 9=9$$
Work Step by Step
$$\lim_{x \to 5} 9=9$$Now, we want to prove this limit by using $\varepsilon - \delta$ definition; that is, we must show that for each $\varepsilon >0$, there exists a $\delta >0$ such that $|9-9|< \varepsilon$ whenever $|x-5|< \delta$.
Now, the inequality $0=|9-9| < \varepsilon$ always holds for any positive real number $\varepsilon$. So, by choosing any positive real number $\delta >0$ (for example, $1$, $\epsilon$ , ...), we conclude that$$|x-5|< \delta \quad \Rightarrow \quad |9-9|=0 < \varepsilon . $$Done.
