Math Problem Statement

Реши задание

Вычислите предел lim (cos x)^(3/x^2), когда x стремится к 0.

Solution

The task is to calculate the limit:

limx0(cosx)3x2.\lim_{x \to 0} (\cos x)^{\frac{3}{x^2}}.

Step-by-step solution:

  1. Rewrite the expression using logarithms:

To simplify, let's take the natural logarithm: y=(cosx)3x2    lny=3x2ln(cosx).y = (\cos x)^{\frac{3}{x^2}} \implies \ln y = \frac{3}{x^2} \ln(\cos x).

The problem now reduces to finding: limx0lny=limx03ln(cosx)x2.\lim_{x \to 0} \ln y = \lim_{x \to 0} \frac{3 \ln(\cos x)}{x^2}.

  1. Expand ln(cosx)\ln(\cos x) using Taylor series:

For small xx, cosx1x22\cos x \approx 1 - \frac{x^2}{2}. 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 -uforsmallfor smallu)}.

So: ln(cosx)x22.\ln(\cos x) \approx -\frac{x^2}{2}.

  1. Substitute back into the limit:

Substituting ln(cosx)x22\ln(\cos x) \approx -\frac{x^2}{2}: limx03ln(cosx)x2limx03(x22)x2=limx032=32.\lim_{x \to 0} \frac{3 \ln(\cos x)}{x^2} \approx \lim_{x \to 0} \frac{3 \cdot \left(-\frac{x^2}{2}\right)}{x^2} = \lim_{x \to 0} \frac{-3}{2} = -\frac{3}{2}.

  1. Exponentiate to find yy:

Since lny=32\ln y = -\frac{3}{2}, we exponentiate both sides to get: y=e32.y = e^{-\frac{3}{2}}.

Final Answer:

The limit is: e32.\boxed{e^{-\frac{3}{2}}}.

Would you like further clarification or details?


Related Questions:

  1. How does the Taylor expansion help in approximating cosx\cos x near zero?
  2. What are the steps to solve a limit of the form af(x)a^{f(x)}?
  3. Why is the natural logarithm used in this problem?
  4. How can L’Hôpital’s rule be applied to logarithmic limits?
  5. What happens if we replace cosx\cos x with sinx\sin x 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