https://file.aiquickdraw.com/mathbot/file/1724861356967gu5vvgd2.jpg

The sequence given in the image is:

\[
u_n = \frac{2n + 1}{n + 325}
\]

To find the limit of this sequence as \( n \) approaches infinity (\( n \to \infty \)), let's analyze it step by step.

### Step 1: Divide Numerator and Denominator by \( n \)
\[
u_n = \frac{2n + 1}{n + 325} = \frac{2n/n + 1/n}{n/n + 325/n} = \frac{2 + \frac{1}{n}}{1 + \frac{325}{n}}
\]

### Step 2: Evaluate the Limit as \( n \to \infty \)
As \( n \) becomes very large, the terms \( \frac{1}{n} \) and \( \frac{325}{n} \) approach 0. So, the expression simplifies to:
\[
\lim_{n \to \infty} u_n = \frac{2 + 0}{1 + 0} = \frac{2}{1} = 2
\]

### Conclusion:
The limit of the sequence \( u_n \) as \( n \) approaches infinity is \( 2 \).

Do you need further details or have any other questions?

Here are five related questions to expand on this topic:
1. How does the limit of a sequence relate to the concept of convergence?
2. Can you apply the same method to find the limit of a different sequence, like \( \frac{3n^2 + n}{n^2 + 5n + 1} \)?
3. What happens if the sequence includes higher powers of \( n \) in the numerator or denominator?
4. How does the presence of constants, like 325 in the denominator, affect the limit?
5. What is the significance of determining the limit of a sequence in mathematical analysis?

**Tip:** When finding the limit of a sequence, dividing both the numerator and the denominator by the highest power of \( n \) often simplifies the expression and makes it easier to determine the limit.

Discover how to find the limit of the sequence u_n = (2n + 1) / (n + 325) as n approaches infinity. This problem explores sequences and limits in mathematics, providing step-by-step analysis and solution.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation