limit as x approaches 4 (x^2-3x-4)^2 over x-4

To solve the limit \(\lim_{{x \to 4}} \frac{(x^2 - 3x - 4)^2}{x - 4}\), we can start by simplifying the expression inside the limit.

### Step 1: Factor the quadratic expression
The expression \(x^2 - 3x - 4\) can be factored as:
\[
x^2 - 3x - 4 = (x - 4)(x + 1)
\]
So the limit becomes:
\[
\lim_{{x \to 4}} \frac{[(x - 4)(x + 1)]^2}{x - 4}
\]

### Step 2: Simplify the expression
Simplifying further, we have:
\[
\lim_{{x \to 4}} \frac{(x - 4)^2(x + 1)^2}{x - 4}
\]
We can cancel one factor of \((x - 4)\) from the numerator and the denominator:
\[
\lim_{{x \to 4}} (x - 4)(x + 1)^2
\]

### Step 3: Apply the limit
Now, we can directly substitute \(x = 4\) into the simplified expression:
\[
(4 - 4)(4 + 1)^2 = 0 \times 25 = 0
\]

Thus, the limit is:
\[
\lim_{{x \to 4}} \frac{(x^2 - 3x - 4)^2}{x - 4} = 0
\]

Would you like more details or have any other questions? 

Here are some related questions you might find interesting:

1. How do you solve limits involving indeterminate forms like \(\frac{0}{0}\)?
2. What is the significance of factoring in limit problems?
3. How do you approach limits at infinity?
4. How do you handle limits involving trigonometric functions?
5. What are some common techniques to solve limits analytically?

**Tip:** Always check if an expression can be factored or simplified before directly substituting values in limit problems. This often avoids indeterminate forms and makes the limit easier to evaluate.

Understand how to solve the limit as x approaches 4 of ((x^2 - 3x - 4)^2) / (x - 4) with a detailed step-by-step solution. This problem involves key concepts in limits, quadratic equations, and factoring, providing insights into analytical approaches to solving limits. Suitable for advanced high school students.

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