The given expression asks for the evaluation of the limit:

$\lim_{{x \to 4}} \frac{\sqrt{40 - x} - 6}{\sqrt{20 - x} - 4}$

Steps to solve:

1. Direct Substitution: If we substitute $x = 4$ directly into the expression, we get:

   $\frac{\sqrt{40 - 4} - 6}{\sqrt{20 - 4} - 4} = \frac{\sqrt{36} - 6}{\sqrt{16} - 4} = \frac{6 - 6}{4 - 4} = \frac{0}{0}$

   Since this gives an indeterminate form $\frac{0}{0}$, we need to use algebraic methods to simplify the expression further.

2. Rationalize the Numerator and Denominator: To remove the square roots, multiply both the numerator and denominator by their respective conjugates.

   Multiply the expression by:

   $\frac{\sqrt{40 - x} + 6}{\sqrt{40 - x} + 6} \times \frac{\sqrt{20 - x} + 4}{\sqrt{20 - x} + 4}$

   This is essentially multiplying by 1, but it helps simplify the expression.

3. Simplify Each Part:

   • Numerator:

     $\left(\sqrt{40 - x} - 6\right)\left(\sqrt{40 - x} + 6\right) = (40 - x) - 36 = 4 - x$

   • Denominator:

     $\left(\sqrt{20 - x} - 4\right)\left(\sqrt{20 - x} + 4\right) = (20 - x) - 16 = 4 - x$

   So the expression simplifies to:

   $\lim_{{x \to 4}} \frac{4 - x}{4 - x} \times \frac{1}{(\sqrt{40 - x} + 6)(\sqrt{20 - x} + 4)}$

   The $4 - x$ terms cancel out, leaving us with:

   $\lim_{{x \to 4}} \frac{1}{(\sqrt{40 - x} + 6)(\sqrt{20 - x} + 4)}$

4. Substitute $x = 4$ Again:

   After cancellation, substitute $x = 4$ into the remaining expression:

   $\frac{1}{(\sqrt{40 - 4} + 6)(\sqrt{20 - 4} + 4)} = \frac{1}{(6 + 6)(4 + 4)} = \frac{1}{12 \times 8} = \frac{1}{96}$

Thus, the value of the limit is:

$\boxed{\frac{1}{96}}$

Would you like more details on any step, or do you have any specific questions?

Here are 5 related questions to consider:

1. How do you approach simplifying expressions involving square roots?
2. What does it mean when a limit results in an indeterminate form?
3. How does multiplying by a conjugate help eliminate square roots?
4. Can you think of another way to simplify this limit using L'Hopital's Rule?
5. How would the result change if the expressions inside the square roots were different?

Tip: Always check for opportunities to cancel terms after rationalizing—it often simplifies the problem greatly.