1. Find the limit of x to -4 on 1/4+1/x over 4+x

To find the limit of the expression \( \frac{\frac{1}{4} + \frac{1}{x}}{4 + x} \) as \( x \to -4 \), let's proceed step by step.

### Step 1: Write the expression
We are dealing with the limit:
\[
\lim_{x \to -4} \frac{\frac{1}{4} + \frac{1}{x}}{4 + x}
\]

### Step 2: Substitute \( x = -4 \) directly
Substituting \( x = -4 \) into the expression:

\[
\frac{\frac{1}{4} + \frac{1}{-4}}{4 - 4} = \frac{\frac{1}{4} - \frac{1}{4}}{0} = \frac{0}{0}
\]

This results in an indeterminate form \( \frac{0}{0} \), so we need to simplify the expression or apply L'Hopital's Rule.

### Step 3: Apply L'Hopital's Rule
Since we have an indeterminate form \( \frac{0}{0} \), we can apply **L'Hopital's Rule**. According to L'Hopital's Rule, we take the derivative of the numerator and denominator separately and then find the limit of the resulting expression.

#### Derivative of the numerator:
\[
\frac{d}{dx} \left( \frac{1}{4} + \frac{1}{x} \right) = 0 - \frac{1}{x^2} = -\frac{1}{x^2}
\]

#### Derivative of the denominator:
\[
\frac{d}{dx} \left( 4 + x \right) = 1
\]

### Step 4: Rewrite the limit
Now we have the new limit:
\[
\lim_{x \to -4} \frac{-\frac{1}{x^2}}{1}
\]
Substituting \( x = -4 \):
\[
-\frac{1}{(-4)^2} = -\frac{1}{16}
\]

### Final Answer:
The limit is:
\[
\lim_{x \to -4} \frac{\frac{1}{4} + \frac{1}{x}}{4 + x} = -\frac{1}{16}
\]

Would you like more details on any step or further clarification?

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Here are 5 related questions to explore:
1. What are the conditions to apply L'Hopital's Rule?
2. How do you find limits that result in indeterminate forms other than \( \frac{0}{0} \)?
3. Can the limit change if we modify the numerator or denominator slightly?
4. How would this limit change if the denominator was \( x - 4 \) instead of \( x + 4 \)?
5. What happens to the limit if \( x \to \infty \) instead of \( x \to -4 \)?

**Tip:** Always check if direct substitution works before using L'Hopital's Rule or other techniques.

Find the limit of x to -4 on 1/4 + 1/x over 4 + x

Learn how to find the limit of (1/4 + 1/x) / (4 + x) as x approaches -4 using L'Hopital's Rule. The problem involves understanding limits, indeterminate forms, and derivatives. This step-by-step solution is ideal for high school students in Grades 11-12 or college-level calculus.

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