The given limit problem is:

$\lim_{{x \to 2}} \frac{{x^2 - 4x + 4}}{{x - 2}}.$

Step-by-step Solution:

1. Simplify the expression:
   First, factor the numerator: $x^2 - 4x + 4 = (x - 2)(x - 2) = (x - 2)^2.$ Thus, the expression becomes: $\frac{{(x - 2)^2}}{{x - 2}}.$

2. Cancel the common term:
   Since $x - 2$ appears in both the numerator and denominator, we can cancel out one of the $x - 2$ terms (as long as $x \neq 2$): $\frac{{(x - 2)^2}}{{x - 2}} = x - 2.$

3. Substitute the limit:
   Now, take the limit as $x \to 2$: $\lim_{{x \to 2}} (x - 2) = 2 - 2 = 0.$

Final Answer:

$\lim_{{x \to 2}} \frac{{x^2 - 4x + 4}}{{x - 2}} = 0.$

Would you like more details on this solution or any other clarifications?

Related Questions:

1. What happens when we try direct substitution in the original limit problem?
2. Can L'Hopital's Rule be applied to this limit, and how would it work?
3. How does factoring help simplify rational expressions?
4. What are some common indeterminate forms in limits?
5. How can we apply limits to real-world problems?

Tip:

Always check for indeterminate forms (like $\frac{0}{0}$) before deciding how to simplify a limit expression.