https://file.aiquickdraw.com/mathbot/file/1726231181584cjxuhj95.jpg

The problem in the image asks to calculate the limit:

\[
\lim_{n \to \infty} \frac{(2n+1) \cdot 25n^4 + 3n^3}{(5n+3) \cdot 52n^4 + 3n^3 + \cos n}
\]

Let's simplify and solve this limit.

### Step 1: Factor the highest power of \(n\) in both the numerator and the denominator.
In the numerator:
\[
(2n+1) \cdot 25n^4 + 3n^3 = 25n^4 \cdot (2n+1) + 3n^3
= 25n^4 \cdot (2n+1) + 3n^3 
\]

In the denominator:
\[
(5n+3) \cdot 52n^4 + 3n^3 + \cos n = 52n^4 \cdot (5n+3) + 3n^3 + \cos n
\]

### Step 2: Simplify by dividing every term by \(n^5\).
This gives us:
\[
\lim_{n \to \infty} \frac{25 \cdot (2 + \frac{1}{n}) + \frac{3}{n^2}}{52 \cdot (5 + \frac{3}{n}) + \frac{3}{n^2} + \frac{\cos n}{n^5}}
\]

### Step 3: Evaluate the limit as \(n \to \infty\).
As \(n\) approaches infinity:
- The terms \(\frac{1}{n}\), \(\frac{1}{n^2}\), and \(\frac{\cos n}{n^5}\) approach 0.
  
Thus, the limit simplifies to:
\[
\frac{25 \cdot 2}{52 \cdot 5} = \frac{50}{260} = \frac{5}{26}
\]

### Conclusion:
The correct limit is \(\boxed{\frac{5}{26}}\).

However, based on the answer choices provided in the image (10, 3, 0.5, and "there is no limit"), none of them match the exact limit calculated. Since none of the provided choices match the calculated limit, it's likely that the correct answer is not provided or "there is no limit" might have been included as a placeholder. 

**Please clarify or ask if you need further assistance.**

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**Relative Questions:**
1. What is the significance of calculating limits in mathematical analysis?
2. How can we apply L'Hôpital's Rule in evaluating limits?
3. When do terms like \(\frac{1}{n}\) and \(\frac{1}{n^2}\) become negligible in limit problems?
4. What are some common pitfalls in solving limit problems?
5. How do trigonometric functions, like \(\cos n\), impact the limit as \(n\) approaches infinity?

**Tip:** Always check whether the problem asks for a specific type of limit, such as a one-sided limit or an infinite limit, as the approach might differ.

Calculate \lim_{n \to \infty} \frac{(2n+1) \cdot 25n^4 + 3n^3}{(5n+3) \cdot 52n^4 + 3n^3 + \cos n}

Learn how to calculate the limit of a complex rational function involving powers of n as it approaches infinity. This problem explores key concepts in limits, including asymptotic behavior and the role of highest degree terms. Suitable for advanced calculus students.

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