Math Problem Statement
Реши задание
Solution
The task is to calculate the limit:
Step-by-step solution:
- Rewrite the expression using logarithms:
To simplify, let's take the natural logarithm:
The problem now reduces to finding:
- Expand using Taylor series:
For small , . Taking the natural logarithm: \ln(\cos x) \approx \ln\left(1 - \frac{x^2}{2}\right) \approx -\frac{x^2}{2} \quad \text{(since \ln(1 - u) \approx -uu)}.
So:
- Substitute back into the limit:
Substituting :
- Exponentiate to find :
Since , we exponentiate both sides to get:
Final Answer:
The limit is:
Would you like further clarification or details?
Related Questions:
- How does the Taylor expansion help in approximating near zero?
- What are the steps to solve a limit of the form ?
- Why is the natural logarithm used in this problem?
- How can L’Hôpital’s rule be applied to logarithmic limits?
- What happens if we replace with in this problem?
Tip:
For problems involving powers of trigonometric functions with small angles, consider logarithmic transformations and series expansions for simplification!
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Math Problem Analysis
Mathematical Concepts
Limits
Exponential Functions
Taylor Series
Logarithms
Formulas
ln(a^b) = b * ln(a)
Taylor expansion: ln(1 - u) ≈ -u (for small u)
cos(x) ≈ 1 - x^2/2 (for small x)
Theorems
Taylor Series Expansion
Properties of Exponents and Logarithms
Suitable Grade Level
Grades 11-12 or University Calculus