For part (e), we are asked to find:

$\lim_{x \to 3} \frac{g(x)}{h(x)}$

From the given information, we know:

$\lim_{x \to 3} g(x) = -2 \quad \text{and} \quad \lim_{x \to 3} h(x) = 0$

Thus, the expression becomes:

$\frac{-2}{0}$

Since dividing by zero leads to an undefined value, the limit does not exist.

Therefore, the answer to part (e) is DNE (Does Not Exist).

Let me know if you'd like further clarification or assistance with other parts of the question!

Here are some related questions to explore:

1. What happens when we divide a non-zero number by zero in the context of limits?
2. How do we handle limits that involve division by zero, particularly when the numerator is non-zero?
3. Can limits involving zero in the denominator ever exist? Under what conditions?
4. What is the concept of an indeterminate form in limits, and how does it apply here?
5. How would you handle limits that involve approaching zero from either the positive or negative direction?

Tip: Always check whether a limit involves division by zero before attempting to solve it!