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The given expression is:

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

### Step-by-step solution:

1. **Simplify the expression inside the limit**:
   We can rewrite the expression as:

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

   The term \(\frac{n^2 + 3}{n}\) can be simplified by distributing the division:

   \[
   \frac{n^2 + 3}{n} = \frac{n^2}{n} + \frac{3}{n} = n + \frac{3}{n}
   \]

   So the expression becomes:

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

2. **Combine like terms**:
   We can now combine the terms:

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

   Rearranging terms:

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

3. **Evaluate the limit**:
   As \(n \to +\infty\):
   - The term \(-n^2\) dominates because it grows faster than \(n\) or \(\frac{3}{n}\).
   - The linear term \(n\) grows slower than \(n^2\), and the term \(\frac{3}{n}\) tends to 0.

   Therefore, the limit is dominated by the \(-n^2\) term:

   \[
   \lim_{n \to +\infty} \left( -n^2 + n + \frac{3}{n} \right) = -\infty
   \]

### Final Answer:

\[
\lim_{n \to +\infty} \left( \frac{n^2 + 3}{n} - n^2 \right) = -\infty
\]

Would you like more details on any step?

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Here are five related questions to expand on this:
1. How do limits help analyze the behavior of functions as they approach infinity?
2. What are the common techniques for solving limits with dominant terms?
3. How do terms with different growth rates affect the overall limit?
4. What happens when you apply L'Hopital's rule to a function with an indeterminate form?
5. Can you give examples of real-world applications of limits in physics or economics?

**Tip**: Always check if one term dominates others in a limit involving infinity—it often simplifies the problem quickly!

Solve the limit as n approaches infinity for the expression: (n^2 + 3) / n - n^2.

This page provides a step-by-step solution to evaluating the limit as n approaches infinity for the expression (n^2 + 3)/n - n^2. Learn about the dominance of terms and how polynomial growth rates affect the outcome. Suitable for high school and college calculus students.

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