limit (sqrt(n^2 + n) - n) as n -> infinity

To evaluate the limit of the expression \( \lim_{n \to \infty} \left(\sqrt{n^2 + n} - n\right) \), let's simplify the expression first.

Given:

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

We can simplify this by multiplying and dividing by the conjugate of the expression:

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

This simplifies to:

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

Now, let's factor out \( n \) from the square root in the denominator:

\[
\lim_{n \to \infty} \frac{n}{n\sqrt{1 + \frac{1}{n}} + n} = \lim_{n \to \infty} \frac{1}{\sqrt{1 + \frac{1}{n}} + 1}
\]

As \( n \) approaches infinity, \( \frac{1}{n} \) approaches 0. Thus, the expression simplifies to:

\[
\lim_{n \to \infty} \frac{1}{\sqrt{1 + 0} + 1} = \frac{1}{1 + 1} = \frac{1}{2}
\]

So, the limit is:

\[
\boxed{\frac{1}{2}}
\]

Would you like any more details or have any questions? 

Here are 5 related questions for further exploration:

1. How would the limit change if the expression inside the square root were \( n^2 + 2n \) instead of \( n^2 + n \)?
2. Can you apply a similar method to evaluate the limit \( \lim_{n \to \infty} \left(\sqrt{n^2 + n^3} - n\right) \)?
3. How does L'Hôpital's rule apply to limits involving square roots?
4. What happens if we consider the limit \( \lim_{n \to \infty} \left(\sqrt{n^2 + k} - n\right) \) for a constant \( k \)?
5. How can the technique of multiplying by the conjugate be used to simplify other types of limits?

**Tip:** When dealing with limits at infinity, multiplying by the conjugate is a powerful technique to simplify square roots or expressions with similar terms.

Learn how to evaluate the limit \( \lim_{n \to \infty} (\sqrt{n^2 + n} - n) \) step-by-step. Understand the properties of limits, infinity, and square roots in this mathematical problem. Suitable for advanced high school students and beyond.

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