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To evaluate the limit 

\[
\lim_{x \to 1} \frac{x^9 - 1}{x^2 - 1},
\]

we can proceed as follows:

### Step 1: Factorize the Expressions
Both the numerator and the denominator can be factorized as differences of powers.

1. **Numerator**: \(x^9 - 1\) is a difference of powers and can be factored as:
   \[
   x^9 - 1 = (x - 1)(x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1).
   \]

2. **Denominator**: \(x^2 - 1\) is a difference of squares and can be factored as:
   \[
   x^2 - 1 = (x - 1)(x + 1).
   \]

### Step 2: Simplify the Expression
Now we substitute the factored forms into the expression:

\[
\frac{x^9 - 1}{x^2 - 1} = \frac{(x - 1)(x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)}{(x - 1)(x + 1)}.
\]

We can cancel out \( (x - 1) \) from both the numerator and the denominator (assuming \( x \neq 1 \)):

\[
= \frac{x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1}{x + 1}.
\]

### Step 3: Substitute \( x = 1 \) into the Simplified Expression
Now, we substitute \( x = 1 \) in the remaining expression:

\[
= \frac{1^8 + 1^7 + 1^6 + 1^5 + 1^4 + 1^3 + 1^2 + 1 + 1}{1 + 1}.
\]

Each power of 1 is simply 1, so we get:

\[
= \frac{1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1}{2} = \frac{9}{2}.
\]

### Final Answer
The limit is:

\[
\lim_{x \to 1} \frac{x^9 - 1}{x^2 - 1} = \frac{9}{2}.
\]

Would you like more details on any step or have further questions?

---

Here are some related questions to expand your understanding:

1. How do you factor expressions involving higher powers, like \(x^n - 1\)?
2. What are other methods to evaluate limits involving indeterminate forms?
3. How does L'Hôpital's Rule apply to limits with \( \frac{0}{0} \) forms?
4. What are some other examples of using the difference of squares in limit problems?
5. How can partial fractions be useful in evaluating limits?

**Tip:** When evaluating limits that result in \( \frac{0}{0} \), try factoring the numerator and denominator first to simplify the expression before applying other techniques.

Evaluate the following limit: 
\[
\lim_{x \to 1} \frac{x^9 - 1}{x^2 - 1}
\]
Enter -I if your answer is -∞, enter I if your answer is ∞, and enter DNE if the limit does not exist.

Explore how to evaluate the limit \( \lim_{x \to 1} \frac{x^9 - 1}{x^2 - 1} \) by factoring and simplifying. Learn key concepts in limits, such as the difference of powers and the difference of squares, in this advanced high school level problem suitable for students in Grades 11-12.

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