simplify x^2 - 16 over x - 4 as x approaches 4

The given expression is:

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

### Step 1: Factor the numerator
The numerator \( x^2 - 16 \) is a difference of squares, so we can factor it as:

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

### Step 2: Simplify the expression
Now the expression becomes:

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

For \( x \neq 4 \), the \( (x - 4) \) terms in the numerator and denominator cancel out:

\[
x + 4
\]

### Step 3: Evaluate as \( x \) approaches 4
Now that we have simplified the expression to \( x + 4 \), we can substitute \( x = 4 \):

\[
4 + 4 = 8
\]

Thus, as \( x \) approaches 4, the value of the expression approaches **8**.

Would you like more details or have any questions?

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Here are 5 related questions to expand your understanding:
1. How do you factor more complex polynomials like \( x^3 - 8 \)?
2. What happens to rational functions when the denominator approaches zero?
3. Can you simplify \( \frac{x^3 - 8}{x - 2} \)?
4. How do you solve limits involving trigonometric functions?
5. What is the difference between continuous and discontinuous functions?

**Tip:** When simplifying rational expressions, always check for factors that can cancel out before evaluating limits.

Learn how to simplify the rational expression (x^2 - 16) / (x - 4) as x approaches 4 by factoring the difference of squares and applying limits. This problem demonstrates key concepts in algebra and calculus, suitable for students in Grades 10-12.

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