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.