https://file.aiquickdraw.com/mathbot/file/17267963062881dlrf85a.jpg

The problem provided is:

\[
\lim_{x \to 4} \frac{x^2 - 16}{x - 4}
\]

To solve this limit, we notice that directly substituting \(x = 4\) leads to an indeterminate form of \(\frac{0}{0}\). This suggests that the expression can be factored or simplified.

### Step 1: Factor the numerator

The numerator \(x^2 - 16\) is a difference of squares:

\[
x^2 - 16 = (x - 4)(x + 4)
\]

### Step 2: Simplify the expression

Substitute the factored form into the original limit expression:

\[
\lim_{x \to 4} \frac{(x - 4)(x + 4)}{x - 4}
\]

Now, cancel out the common factor \(x - 4\) from the numerator and denominator:

\[
\lim_{x \to 4} (x + 4)
\]

### Step 3: Evaluate the limit

Now that the expression is simplified, substitute \(x = 4\):

\[
4 + 4 = 8
\]

Thus, the value of the limit is:

\[
\boxed{8}
\]

Would you like more details, or do you have any further questions?

### Related Questions:
1. What would the limit be if the expression were \( \lim_{x \to -4} \frac{x^2 - 16}{x + 4} \)?
2. How do you apply L'Hopital's Rule to solve indeterminate forms like \( \frac{0}{0} \)?
3. What is the significance of factoring in solving limits algebraically?
4. Can this type of limit be solved using a graphical approach? If so, how?
5. What are the conditions under which we cannot cancel terms directly in limit problems?

### Tip:
Always check for common factors in the numerator and denominator when encountering indeterminate forms such as \( \frac{0}{0} \). This often simplifies the limit problem.

Find the limit of (x^2 - 16)/(x - 4) as x approaches 4. This problem involves factoring a quadratic expression to resolve an indeterminate form. It explores key algebraic concepts and limit laws, making it suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation