The image contains a mathematical problem involving a limit calculation. I will analyze and provide the solution. Let's first transcribe the given limit:

Problem

$\lim_{n \to \infty} \frac{n^2 + 2n + 1}{2n^2 + 3n + 3}.$

Solution

1. Identify the dominant term in the numerator and denominator:

   • Numerator: $n^2 + 2n + 1$, where the dominant term is $n^2$.
   • Denominator: $2n^2 + 3n + 3$, where the dominant term is $2n^2$.

2. Divide all terms by $n^2$ (the highest power of $n$):

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

3. Simplify each term:

   • Numerator: $\frac{n^2}{n^2} = 1$, $\frac{2n}{n^2} = \frac{2}{n}$, and $\frac{1}{n^2} = \frac{1}{n^2}$.
   • Denominator: $\frac{2n^2}{n^2} = 2$, $\frac{3n}{n^2} = \frac{3}{n}$, and $\frac{3}{n^2} = \frac{3}{n^2}$.

   After substitution, the expression becomes:

   $\frac{1 + \frac{2}{n} + \frac{1}{n^2}}{2 + \frac{3}{n} + \frac{3}{n^2}}.$

4. Take the limit as $n \to \infty$:

   • Terms involving $\frac{1}{n}$ or $\frac{1}{n^2}$ approach $0$.
   • Thus, the expression simplifies to:

   $\frac{1 + 0 + 0}{2 + 0 + 0} = \frac{1}{2}.$

Final Answer:

$\boxed{\frac{1}{2}}$

Would you like a deeper explanation or visualization? Let me know!

Here are 5 related questions to explore further:

1. How does the behavior of a rational function change as $n \to \infty$?
2. What happens when the numerator and denominator have the same degree?
3. What if the denominator has a higher degree than the numerator?
4. How can L’Hôpital’s Rule be applied to similar limits?
5. Why do we ignore lower-order terms as $n \to \infty$?

Tip: Always divide by the highest power of $n$ to simplify limits involving polynomials.