The given limit is:

$\lim_{n \to \infty} \left( \frac{3n + 3}{n + 2} - \frac{2n + 2}{2n + 4} \right)^2$

Let's break this down step by step:

Step 1: Simplify each term inside the parentheses as $n \to \infty$.

• For the first term: $\frac{3n + 3}{n + 2}$, factor out $n$ from the numerator and the denominator:

$\frac{3n + 3}{n + 2} = \frac{n(3 + \frac{3}{n})}{n(1 + \frac{2}{n})} = \frac{3 + \frac{3}{n}}{1 + \frac{2}{n}}.$

As $n \to \infty$, both $\frac{3}{n}$ and $\frac{2}{n}$ tend to zero, so this term tends to $\frac{3}{1} = 3$.

• For the second term: $\frac{2n + 2}{2n + 4}$, factor out $n$ from the numerator and denominator:

$\frac{2n + 2}{2n + 4} = \frac{n(2 + \frac{2}{n})}{n(2 + \frac{4}{n})} = \frac{2 + \frac{2}{n}}{2 + \frac{4}{n}}.$

As $n \to \infty$, both $\frac{2}{n}$ and $\frac{4}{n}$ tend to zero, so this term tends to $\frac{2}{2} = 1$.

Step 2: Subtract the two simplified terms.

Now we subtract the two terms:

$\lim_{n \to \infty} \left( 3 - 1 \right)^2 = (2)^2 = 4.$

Final Answer:

The value of the limit is $\boxed{4}$.

Would you like more details on any specific step or have any other questions?

Follow-up Questions:

1. How would the solution change if the powers inside the limit were different (e.g., cube instead of square)?
2. What happens if the terms in the denominator grow faster than the numerator as $n \to \infty$?
3. Can L'Hopital's rule be applied to simplify this limit problem?
4. How do limits at infinity differ from limits at finite points?
5. What is the significance of factoring out $n$ when dealing with limits involving large values of $n$?

Tip:

When simplifying limits involving infinity, try to factor out the highest power of $n$ to see the leading behavior of the expression.