To evaluate the limit

$\lim_{n \to \infty} \sqrt{n^2 + 2} - n$

we'll proceed by rationalizing the expression.

Step 1: Rewrite the Expression

Rewrite the expression by multiplying the numerator and denominator by the conjugate of the square root term:

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

This simplifies to:

$\lim_{n \to \infty} \frac{(n^2 + 2) - n^2}{\sqrt{n^2 + 2} + n}$

Step 2: Simplify the Expression

The $n^2$ terms in the numerator cancel each other out:

$\lim_{n \to \infty} \frac{2}{\sqrt{n^2 + 2} + n}$

Step 3: Analyze the Denominator

For large $n$, the expression $\sqrt{n^2 + 2}$ is approximately $n$, so the denominator becomes approximately $n + n = 2n$:

$\lim_{n \to \infty} \frac{2}{2n} = \lim_{n \to \infty} \frac{1}{n}$

Step 4: Take the Limit

As $n \to \infty$, $\frac{1}{n} \to 0$.

Final Answer

$\lim_{n \to \infty} \sqrt{n^2 + 2} - n = 0$

Would you like more details on any of these steps?

Here are some related questions:

1. How does rationalizing help in evaluating limits involving square roots?
2. What are other techniques for evaluating limits that approach infinity?
3. Why does $\sqrt{n^2 + 2}$ approximate $n$ as $n \to \infty$?
4. How would the solution change if we had a different constant added, such as $\sqrt{n^2 + k}$?
5. What are the limitations of using conjugates to simplify expressions?

Tip: Rationalizing is a powerful tool for simplifying limits with square roots, especially when they approach infinity!