We are asked to evaluate the following limit in terms of the constant $a$:

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

Step 1: Factor the expression

1. The numerator is a difference of squares: $x^2 - a^2 = (x - a)(x + a)$

2. The denominator can be factored as well. Notice it is a quadratic in $x$, so we attempt to factor it as: $x^2 - 3ax + 2a^2 = (x - 2a)(x - a)$

Step 2: Substitute the factored forms

We now rewrite the limit with the factored forms of the numerator and denominator:

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

Step 3: Simplify the expression

The term $(x - a)$ appears in both the numerator and the denominator, so we can cancel it, assuming $x \neq a$ (which is valid as we are taking the limit):

$\lim_{x \to a} \frac{x + a}{x - 2a}$

Step 4: Evaluate the limit

Now, substitute $x = a$ into the simplified expression:

$\frac{a + a}{a - 2a} = \frac{2a}{-a} = -2$

Final Answer:

The value of the limit is:

$\boxed{-2}$

Let me know if you would like any further details, or if you have any questions! Here are a few related questions you might find useful:

1. How do you factor more complex quadratic expressions?
2. What are the common techniques for simplifying limits?
3. How do you apply L'Hopital's Rule in similar cases?
4. When is it appropriate to cancel terms in a limit expression?
5. How do you evaluate limits where indeterminate forms like $\frac{0}{0}$ arise?

Tip: Always try factoring before applying more advanced methods for limit evaluation, as it often simplifies the expression quickly!