https://file.aiquickdraw.com/mathbot/file/1724877352088zq18mtn6.jpg

The sequence \( u_n \) given in the image is defined by:

\[
u_n = \sqrt{n^2 + n - n}
\]

Simplifying the expression inside the square root, we get:

\[
u_n = \sqrt{n^2}
\]

Since \( \sqrt{n^2} = |n| \), and for positive \( n \) (which is typical when dealing with sequences indexed by \( n \)), \( |n| = n \). Therefore, we have:

\[
u_n = n
\]

To find the limit of this sequence as \( n \to \infty \), we observe that the sequence \( u_n = n \) simply increases without bound as \( n \) increases.

Thus, the limit of the sequence \( u_n \) as \( n \to \infty \) is:

\[
\lim_{n \to \infty} u_n = \infty
\]

In summary, the sequence \( u_n = \sqrt{n^2 + n - n} \) diverges to infinity as \( n \to \infty \).

Would you like more details or have any specific questions?

### Related Questions:
1. What would be the limit if the sequence was \( u_n = \sqrt{n^2 + n + 1} \)?
2. How would the limit change if the sequence were \( u_n = \sqrt{n^2 - n} \)?
3. What happens to the limit if \( n \) is replaced by \( -n \) in the sequence \( u_n = \sqrt{n^2 + n - n} \)?
4. Can we determine the rate at which \( u_n = \sqrt{n^2 + n - n} \) diverges as \( n \to \infty \)?
5. What is the general behavior of a sequence defined by \( u_n = \sqrt{n^2 + an + b} \) where \( a \) and \( b \) are constants?

### Tip:
When dealing with square roots in sequences, it’s often useful to simplify the expression inside the root first and then consider the asymptotic behavior as \( n \) becomes large.

Explore the mathematical problem of finding the limit of the sequence defined by u_n = √(n² + n - n) as n approaches infinity. Learn how to simplify the sequence and determine its behavior. Suitable for advanced high school students and anyone interested in mathematical sequences.

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