The problem asks us to evaluate the following limit as $x$ approaches 2 from the right:

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

Step 1: Factor the expression

Numerator:

The numerator is $x^2 - 2x$, which can be factored as:

$x(x - 2)$

Denominator:

The denominator is $x^2 - 4x + 4$, which can be factored as:

$(x - 2)^2$

Step 2: Simplify the expression

Now, rewrite the limit using these factored forms:

$\frac{x(x - 2)}{(x - 2)^2}$

Cancel out one $(x - 2)$ term from the numerator and the denominator:

$\frac{x}{x - 2}$

Step 3: Evaluate the limit as $x \to 2^+$

Now we need to evaluate the limit of $\frac{x}{x - 2}$ as $x$ approaches 2 from the right ($2^+$):

As $x \to 2^+$, the term $x - 2$ becomes a small positive number. Thus, the expression behaves like:

$\frac{2}{\text{small positive number}} \to +\infty$

Conclusion:

The limit is $+\infty$, which corresponds to option d. ∞.

Would you like further details or explanations?

Here are 5 related questions to explore:

1. What is the difference between evaluating limits from the right and from the left?
2. How do you handle limits when the numerator and denominator both tend to 0 (indeterminate forms)?
3. What is L'Hopital's Rule, and when should it be applied in limits?
4. Can limits always be simplified by factoring, or are there cases when another method is more suitable?
5. What are some typical limits that approach infinity, and how do we interpret them?

Tip: Always check for indeterminate forms like $\frac{0}{0}$ when solving limits, as they often indicate the need for factoring or applying other limit techniques like L'Hopital's Rule.