https://file.aiquickdraw.com/mathbot/file/1727691949358ggvjswnn.jpg

The limit problem provided is:

\[
\lim_{x \to 0} \frac{1 - \cos(x)}{5x^2}
\]

To evaluate this limit, we will use a known limit result:

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

Using this result, we can simplify the given expression:

\[
\lim_{x \to 0} \frac{1 - \cos(x)}{5x^2} = \frac{1}{5} \cdot \lim_{x \to 0} \frac{1 - \cos(x)}{x^2} = \frac{1}{5} \cdot \frac{1}{2} = \frac{1}{10}
\]

Thus, the value of the limit is:

\[
\boxed{\frac{1}{10}}
\]

Would you like more details, or do you have any questions?

Here are 5 related questions to consider:
1. How do you apply L'Hopital's Rule to similar limit problems?
2. What is the Taylor series expansion for \(\cos(x)\) and how can it help in solving limits?
3. How does the small-angle approximation for \(\cos(x)\) help in simplifying limits?
4. What other limits are useful to memorize for trigonometric functions?
5. How would this limit change if the function involved \(\sin(x)\) instead of \(\cos(x)\)?

**Tip:** The Taylor series expansion of a function can often simplify evaluating limits, especially for trigonometric functions.

Evaluate the limit: lim(x -> 0) [1 - cos(x)] / 5x^2

Learn how to evaluate the limit lim(x -> 0) [1 - cos(x)] / 5x^2. This problem involves the use of trigonometric limit rules and can be solved using L'Hopital's Rule or Taylor series expansion. Ideal for students in Grades 11-12 studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation