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

To simplify the expression \(\frac{x^2 - 16}{x - 4}\) as \(x\) approaches 4, we follow these steps:

### 1. Factor the numerator
Notice that \(x^2 - 16\) is a difference of squares, which can be factored as:
\[
x^2 - 16 = (x - 4)(x + 4)
\]

### 2. Rewrite the expression
Now, rewrite the original expression \(\frac{x^2 - 16}{x - 4}\) using the factored form of the numerator:
\[
\frac{x^2 - 16}{x - 4} = \frac{(x - 4)(x + 4)}{x - 4}
\]

### 3. Simplify by canceling common factors
Since \(x - 4\) is a factor in both the numerator and the denominator (and \(x \neq 4\) to avoid division by zero), you can cancel out the \(x - 4\) terms:
\[
\frac{(x - 4)(x + 4)}{x - 4} = x + 4
\]

### 4. Evaluate the limit as \(x \to 4\)
Now, as \(x\) approaches 4, the simplified expression becomes:
\[
x + 4
\]
Substitute \(x = 4\):
\[
4 + 4 = 8
\]

### Final Answer:
\[
\lim_{x \to 4} \frac{x^2 - 16}{x - 4} = 8
\]

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. What is the difference of squares and how is it applied in factoring?
2. Why does the \(x - 4\) term cancel out, and what does it imply about limits?
3. How do you evaluate limits involving removable discontinuities?
4. What happens if \(x\) were exactly equal to 4 in the original expression?
5. How does this technique apply to more complex rational expressions?

**Tip:** When simplifying rational expressions, always check for common factors that can be canceled before substituting values in limits.

Learn how to simplify the rational expression x^2 - 16 over x - 4 as x approaches 4, using factoring techniques and limits. This step-by-step guide explains the difference of squares and evaluates the limit for this algebraic problem. Suitable for students in Grades 9-12.

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