To evaluate the limit of the expression $\lim_{x \to 5} \frac{f(x) + g(x)}{6f(x)}$, we can apply the properties of limits, which allow us to evaluate the limit of the numerator and denominator separately.

Given: $\lim_{x \to 5} f(x) = 6 \quad \text{and} \quad \lim_{x \to 5} g(x) = 4$

Now, applying the limits to the expression:

$\lim_{x \to 5} \frac{f(x) + g(x)}{6f(x)} = \frac{\lim_{x \to 5} (f(x) + g(x))}{\lim_{x \to 5} 6f(x)}$

Step 1: Evaluate the numerator:

$\lim_{x \to 5} (f(x) + g(x)) = \lim_{x \to 5} f(x) + \lim_{x \to 5} g(x) = 6 + 4 = 10$

Step 2: Evaluate the denominator:

$\lim_{x \to 5} 6f(x) = 6 \cdot \lim_{x \to 5} f(x) = 6 \cdot 6 = 36$

Step 3: Combine the results:

$\frac{10}{36} = \frac{5}{18}$

Thus, the limit is: $\boxed{\frac{5}{18}}$

Would you like further details on any part of the process?

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Here are 5 related questions to explore:

1. How do you apply limit laws to more complex expressions?
2. What happens if a limit involves indeterminate forms, like $\frac{0}{0}$?
3. How do you handle limits involving infinity?
4. Can L'Hopital's Rule be applied to this problem, and why or why not?
5. What if the limits for $f(x)$ and $g(x)$ approached infinity?

Tip: Always check if the limit form is indeterminate before deciding on an evaluation method.