[5 points] Evaluate the following limit in terms of the constant a.
limx→a. (x2 − a2)/(x^2 − 3ax + 2a^2)

We are tasked with evaluating the limit:

\[
\lim_{x \to a} \frac{x^2 - a^2}{x^2 - 3ax + 2a^2}
\]

### Step 1: Factorize the expression

#### Numerator:
The numerator \( x^2 - a^2 \) is a difference of squares, which can be factorized as:

\[
x^2 - a^2 = (x - a)(x + a)
\]

#### Denominator:
The denominator \( x^2 - 3ax + 2a^2 \) can be factorized by recognizing it as a quadratic expression. We need to find factors of \( x^2 - 3ax + 2a^2 \).

Factoring gives:

\[
x^2 - 3ax + 2a^2 = (x - a)(x - 2a)
\]

### Step 2: Simplify the expression
Now we can rewrite the original limit as:

\[
\lim_{x \to a} \frac{(x - a)(x + a)}{(x - a)(x - 2a)}
\]

For \( x \neq a \), we can cancel out the common factor \( (x - a) \):

\[
\lim_{x \to a} \frac{x + a}{x - 2a}
\]

### Step 3: Substitute \( x = a \)
Now, substitute \( x = a \) into the simplified expression:

\[
\frac{a + a}{a - 2a} = \frac{2a}{-a} = -2
\]

Thus, the value of the limit is:

\[
\boxed{-2}
\]

Let me know if you want further details or explanations.

Here are 5 related questions:
1. What happens if the limit involves a different factorization pattern in the denominator?
2. How would the result change if \( a = 0 \)?
3. Can L'Hopital's rule be applied in this situation? Why or why not?
4. How can limits be used to determine continuity of a function?
5. How do limits of rational functions behave near vertical asymptotes?

**Tip**: When solving limits involving rational expressions, always look for opportunities to factorize and simplify before direct substitution.

[5 points] Evaluate the following limit in terms of the constant a.
lim_{x→a} \frac{x^2 − a^2}{x^2 − 3ax + 2a^2}

Learn how to evaluate the limit lim_{x→a} (x^2 − a^2)/(x^2 − 3ax + 2a^2) by factoring algebraic expressions. This solution involves factoring the numerator and denominator, canceling terms, and simplifying the expression to find the limit value. Ideal for Grades 10-12.

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