Math Problem Statement
Final answer just
Calculate \lim_{n \to \infty} \frac{(2n+1) \cdot 25n^4 + 3n^3}{(5n+3) \cdot 52n^4 + 3n^3 + \cos n}
Solution
The problem in the image asks to calculate the limit:
Let's simplify and solve this limit.
Step 1: Factor the highest power of in both the numerator and the denominator.
In the numerator:
= 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}}$$. However, based on the answer choices provided in the image (10, 3, 0.5, and "there is no limit"), none of them match the exact limit calculated. Since none of the provided choices match the calculated limit, it's likely that the correct answer is not provided or "there is no limit" might have been included as a placeholder. **Please clarify or ask if you need further assistance.** --- **Relative Questions:** 1. What is the significance of calculating limits in mathematical analysis? 2. How can we apply L'Hôpital's Rule in evaluating limits? 3. When do terms like $$\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Calculus
Asymptotic Behavior
Formulas
Limit Definition
Highest Degree Terms
Theorems
Limit Theorem
Asymptotic Analysis
Suitable Grade Level
University/Advanced Calculus
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