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Evaluate the limit by using algebric techniques 

The given problem is:

\[
\lim_{x \to a} \frac{x^n - a^n}{x^m - a^m}
\]

We will solve this limit using algebraic techniques.

### Step-by-Step Solution

1. **Factorization**:
   Both the numerator and the denominator can be factored using the difference of powers formula. For \(x^n - a^n\) and \(x^m - a^m\), we use the following factorizations:

   \[
   x^n - a^n = (x - a)(x^{n-1} + x^{n-2}a + \cdots + a^{n-1})
   \]
   
   Similarly,
   
   \[
   x^m - a^m = (x - a)(x^{m-1} + x^{m-2}a + \cdots + a^{m-1})
   \]

2. **Substitute the Factorized Form**:

   After factorizing both the numerator and the denominator, we substitute them into the limit expression:

   \[
   \lim_{x \to a} \frac{(x - a)(x^{n-1} + x^{n-2}a + \cdots + a^{n-1})}{(x - a)(x^{m-1} + x^{m-2}a + \cdots + a^{m-1})}
   \]

3. **Cancel Common Terms**:
   The \((x - a)\) terms cancel out, leaving us with the following expression:

   \[
   \lim_{x \to a} \frac{x^{n-1} + x^{n-2}a + \cdots + a^{n-1}}{x^{m-1} + x^{m-2}a + \cdots + a^{m-1}}
   \]

4. **Evaluate the Limit**:
   Now, we can substitute \(x = a\) directly into the remaining expression:

   \[
   \frac{a^{n-1} + a^{n-1} + \cdots + a^{n-1}}{a^{m-1} + a^{m-1} + \cdots + a^{m-1}}
   \]

   Both the numerator and denominator consist of sums of terms involving powers of \(a\). The numerator has \(n\) terms, each equal to \(a^{n-1}\), so it simplifies to \(n \cdot a^{n-1}\). Similarly, the denominator has \(m\) terms, each equal to \(a^{m-1}\), so it simplifies to \(m \cdot a^{m-1}\).

5. **Final Expression**:

   The limit becomes:

   \[
   \frac{n \cdot a^{n-1}}{m \cdot a^{m-1}}
   \]

6. **Simplify**:
   This simplifies further to:

   \[
   \frac{n}{m} \cdot a^{n-m}
   \]

### Final Answer:

\[
\lim_{x \to a} \frac{x^n - a^n}{x^m - a^m} = \frac{n}{m} \cdot a^{n-m}
\]

Let me know if you would like further clarifications.

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Here are five questions that build on this:

1. What happens if \(n = m\) in the given limit?
2. How would you approach this limit if \(a = 0\)?
3. Can you explain the difference of powers formula and how it's used in limits?
4. How does the limit change if we were working with different variables, such as \(x^p - a^p\) and \(x^q - a^q\)?
5. Could L'Hôpital's rule be used as an alternative method for solving this limit?

**Tip**: When factoring polynomials involving powers, always look for patterns like the difference of squares or sums of powers to simplify the expressions!

Learn how to evaluate the limit \( \lim_{x \to a} \frac{x^n - a^n}{x^m - a^m} \) using algebraic techniques. This problem involves factoring using the difference of powers formula and simplifying the expression to find the final answer. Suitable for advanced high school and college students.

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