We are asked to estimate the limit:

$\lim_{x \to 4} \left( \frac{x + 4}{\sqrt{5 - x} - 3} \right)$

Step-by-Step Approach:

1. Substitute $x = 4$ directly: $\text{Numerator: } x + 4 = 4 + 4 = 8$ $\text{Denominator: } \sqrt{5 - x} - 3 = \sqrt{5 - 4} - 3 = \sqrt{1} - 3 = 1 - 3 = -2$

2. Compute the expression: $\frac{8}{-2} = -4$

Thus, the estimated value of the limit is:

$\boxed{-4}$

Would you like further details or clarification on this process? Here are a few related questions that might expand your understanding:

1. What would happen if the denominator turned out to be zero? How would we deal with that?
2. What is the general technique for evaluating limits that involve square roots?
3. Could L'Hopital's Rule be applied if we had a $\frac{0}{0}$ indeterminate form here?
4. How would we evaluate this limit graphically by looking at the behavior of the function near $x = 4$?
5. How do we estimate limits numerically if direct substitution is difficult?

Tip: Always check for indeterminate forms before directly substituting values in limit problems.