Using: limx→5f(x)=6 and limx→5g(x)=4, evaluate
limx→5f(x)+g(x)6f(x).
Limit =

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.

Using: limx→5f(x)=6 and limx→5g(x)=4, evaluate limx→5(f(x) + g(x)) / 6f(x). Limit =

Learn how to evaluate the limit of (f(x) + g(x)) / 6f(x) as x approaches 5 using basic limit laws. This step-by-step solution involves applying the limit to both the numerator and denominator separately, suitable for students in Grades 11-12 studying Calculus.

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