We are tasked with finding the limit:

$\lim_{n \to \infty} \left( n + \frac{1}{n} - \frac{n^2}{n+1} \right)$

Step-by-step Solution:

1. Simplify the expression: $\lim_{n \to \infty} \left( n + \frac{1}{n} - \frac{n^2}{n+1} \right)$

   The key is to handle each term separately.

2. Break the expression into parts:

   $n + \frac{1}{n} - \frac{n^2}{n+1}$

   Let's simplify the last term $\frac{n^2}{n+1}$.

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

   As $n \to \infty$, the term $\frac{1}{n} \to 0$, so we can approximate the last term as:

   $\frac{n}{1 + \frac{1}{n}} \approx n$

3. Rewriting the expression: Now, the expression becomes: $\lim_{n \to \infty} \left( n + \frac{1}{n} - n \right)$

4. Simplify further: $n - n + \frac{1}{n} = \frac{1}{n}$

5. Find the limit: $\lim_{n \to \infty} \frac{1}{n} = 0$

Thus, the limit is:

$\boxed{0}$

Would you like additional clarification or details on any part of the process?

Here are 5 related questions to explore further:

1. How do we handle limits that involve polynomials and rational expressions?
2. What happens when dealing with limits approaching infinity for more complex functions?
3. Can L'Hopital's Rule be used to solve this limit? Why or why not?
4. What are the differences between solving limits analytically and using approximations?
5. How do higher-order terms in a series expansion affect limits as $n \to \infty$?

Tip: When dealing with rational functions in limits approaching infinity, simplifying by dividing terms and focusing on dominant powers is often the most effective strategy!