Math Problem Statement

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

Solution

To find the limit of the expression 14+1x4+x\frac{\frac{1}{4} + \frac{1}{x}}{4 + x} as x4x \to -4, let's proceed step by step.

Step 1: Write the expression

We are dealing with the limit: limx414+1x4+x\lim_{x \to -4} \frac{\frac{1}{4} + \frac{1}{x}}{4 + x}

Step 2: Substitute x=4x = -4 directly

Substituting x=4x = -4 into the expression:

14+1444=14140=00\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 00\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 00\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:

ddx(14+1x)=01x2=1x2\frac{d}{dx} \left( \frac{1}{4} + \frac{1}{x} \right) = 0 - \frac{1}{x^2} = -\frac{1}{x^2}

Derivative of the denominator:

ddx(4+x)=1\frac{d}{dx} \left( 4 + x \right) = 1

Step 4: Rewrite the limit

Now we have the new limit: limx41x21\lim_{x \to -4} \frac{-\frac{1}{x^2}}{1} Substituting x=4x = -4: 1(4)2=116-\frac{1}{(-4)^2} = -\frac{1}{16}

Final Answer:

The limit is: limx414+1x4+x=116\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?


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 00\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 x4x - 4 instead of x+4x + 4?
  5. What happens to the limit if xx \to \infty instead of x4x \to -4?

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

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Limits
Indeterminate Forms
L'Hopital's Rule

Formulas

L'Hopital's Rule
Derivative of 1/x = -1/x^2

Theorems

L'Hopital's Rule

Suitable Grade Level

Grades 11-12 or College Calculus