We are tasked with evaluating the limit:

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

### Step-by-Step Solution

#### Step 1: Break down the expression
The given limit consists of two terms:

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

We will evaluate each part separately.

#### Step 2: First term - \( \frac{n}{n^2 + 3} \)
For large values of \(n\), the \(n^2\) term dominates over the constant 3 in the denominator. Therefore, we can approximate:

\[
\frac{n}{n^2 + 3} \approx \frac{n}{n^2} = \frac{1}{n}
\]

As \(n \to \infty\), \( \frac{1}{n} \to 0 \). Therefore, the limit of this term is:

\[
\lim_{n \to \infty} \frac{n}{n^2 + 3} = 0
\]

#### Step 3: Second term - \( n^2 \)
As \(n \to \infty\), the second term \( n^2 \) grows without bound:

\[
\lim_{n \to \infty} n^2 = \infty
\]

#### Step 4: Combine the results
Now, combining the limits:

\[
\lim_{n \to \infty} \left( \frac{n}{n^2 + 3} - n^2 \right) = 0 - \infty = -\infty
\]

### Conclusion:
The limit is:

\[
\boxed{-\infty}
\]

Let me know if you would like more details or have any further questions!

---

1. How does \( n^2 \) dominate over other terms as \(n\) increases?
2. What happens if we add a constant to \( n^2 \) in the denominator?
3. How does the approximation \( \frac{1}{n} \) become valid for large \(n\)?
4. Why does the limit of \(n^2\) tend to infinity as \(n \to \infty\)?
5. What are other common techniques to evaluate limits as \(n \to \infty\)?

**Tip**: When evaluating limits, always check if dominant terms simplify the expression!

Explore the step-by-step solution for evaluating the limit n/(n^2 + 3) − n^2 as n tends to infinity. Learn about key mathematical concepts like limits, asymptotic behavior, and the dominance of higher-degree terms for large values of n. Suitable for students in Grades 11-12.

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