Why can't we choose a $\delta$ that depends on $x$?

Question

For example, when asked to prove

$$\lim_{x \to 5}(x^2-9) = 16$$

We see that

$$|f(x)-L| = |x-5||x+5|$$

My question is, why can't we set

$$\delta = \frac{\epsilon}{|x+5|}$$

so that

$$|x-5||x+5|<|x+5| \delta$$ $$\implies |x-5||x+5|<|x+5| \cdot \frac{\epsilon}{|x+5|}$$ $$\implies |x-5||x+5|<\epsilon$$

Answer 1

Because $\delta$ is the bound that tells us how far away from $5$ we're allowed to choose our $x$. In other words, $\delta$ must be chosen before we can pick any $x$, and therefore cannot depend on it.