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.