We are tasked with evaluating the limit:

\[
\lim_{x \to 0} \frac{\log(\cos x)}{\sin^2 x}
\]

### Step 1: Series expansions for small \( x \)

For small \( x \), we can use the Taylor series expansions of \( \cos x \), \( \sin x \), and \( \log(1 + u) \) around 0.

- \( \cos x \approx 1 - \frac{x^2}{2} \) (up to second order in \( x \))
- \( \sin x \approx x - \frac{x^3}{6} \) (up to third order in \( x \))

Using these approximations, we now expand \( \log(\cos x) \) around 0.

\[
\log(\cos x) = \log\left(1 - \frac{x^2}{2}\right) \approx -\frac{x^2}{2}
\]

So, for small \( x \), \( \log(\cos x) \approx -\frac{x^2}{2} \).

### Step 2: Approximate the denominator

For small \( x \), the expansion of \( \sin x \) gives:

\[
\sin^2 x \approx x^2
\]

### Step 3: Substitute approximations into the limit

Now, we substitute the approximations into the original limit:

\[
\lim_{x \to 0} \frac{-\frac{x^2}{2}}{x^2} = \lim_{x \to 0} -\frac{1}{2} = -\frac{1}{2}
\]

### Final Answer:
\[
\lim_{x \to 0} \frac{\log(\cos x)}{\sin^2 x} = -\frac{1}{2}
\]

Would you like more details or have any questions?

Here are five related questions to explore further:
1. How do you derive the Taylor series of a function like \( \cos x \)?
2. What other methods can be used to evaluate limits involving logarithms and trigonometric functions?
3. How does L'Hopital's rule apply to limits like this?
4. Can this limit be solved using a different approximation technique?
5. How does the result change for different trigonometric or logarithmic functions?

**Tip:** When evaluating limits with trigonometric functions, using Taylor series expansions around critical points (like 0) is often very effective for simplification.

Explore the solution to the limit of log(cos x) divided by sin^2 x as x approaches 0. This calculus problem uses Taylor series expansions and logarithmic approximations to simplify the expression. Learn how these methods lead to a final answer of -1/2, suitable for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation