To find the limit of the sequence $u_n$ as $n$ approaches infinity, we start with the given expression:

$u_n = \frac{2n^2 - 3n + 2}{1 - n}$

Step 1: Simplify the expression

We can begin by dividing both the numerator and the denominator by $n^2$, which is the highest power of $n$ in the numerator:

$u_n = \frac{\frac{2n^2}{n^2} - \frac{3n}{n^2} + \frac{2}{n^2}}{\frac{1}{n^2} - \frac{n}{n^2}} = \frac{2 - \frac{3}{n} + \frac{2}{n^2}}{\frac{1}{n^2} - \frac{1}{n}}$

Step 2: Analyze the limit as $n$ approaches infinity

As $n$ becomes very large:

• $\frac{3}{n}$ and $\frac{2}{n^2}$ in the numerator both approach 0.
• $\frac{1}{n^2}$ in both the numerator and the denominator approaches 0.
• $\frac{1}{n}$ in the denominator approaches 0 as well.

This simplifies the expression to:

$u_n \approx \frac{2 + 0 + 0}{0 - 1} = \frac{2}{-1} = -2$

Conclusion

The limit of the sequence $u_n$ as $n$ approaches infinity is $-2$.

Would you like more details or have any other questions?

Here are some related questions you might find interesting:

1. How does the behavior of the sequence change if we modify the coefficients?
2. What happens to the limit if the degree of the numerator and denominator are the same?
3. Can you explain how L'Hôpital's rule could be used to find the limit in a different way?
4. What if the numerator and denominator were polynomials of different degrees?
5. How would the limit change if the sequence had a different highest power of $n$?

Tip: When analyzing limits of sequences or functions, simplifying by dividing by the highest power of $n$ is a very effective technique!