https://file.aiquickdraw.com/mathbot/file/1725776772417xjyxtx8x.jpg

The limit to solve is:

\[
\lim_{x \to 0} \frac{x^{1000} \sin(5x)}{\sin(999x)}
\]

To solve this limit, we can use the small-angle approximation for sine, where \(\sin(\theta) \approx \theta\) as \(\theta \to 0\). Since \(x \to 0\), we can approximate:

\[
\sin(5x) \approx 5x \quad \text{and} \quad \sin(999x) \approx 999x
\]

Substituting these approximations into the limit gives:

\[
\lim_{x \to 0} \frac{x^{1000} \cdot 5x}{999x} = \lim_{x \to 0} \frac{5x^{1001}}{999x} = \lim_{x \to 0} \frac{5x^{1000}}{999}
\]

Since \(x^{1000} \to 0\) as \(x \to 0\), the limit evaluates to:

\[
\frac{5 \cdot 0}{999} = 0
\]

Thus, the final answer is:

\[
\boxed{0}
\]

Would you like more details or have any further questions? Here are some related questions you might find helpful:

1. How does the small-angle approximation work for trigonometric functions?
2. What are other methods to solve limits involving sine and cosine functions?
3. How would the limit change if the powers of \(x\) were different?
4. How does L'Hôpital's Rule apply to this type of limit problem?
5. What happens if we consider the higher-order terms in the sine approximation?

**Tip:** Always check if the small-angle approximation is valid when dealing with trigonometric limits where \(x\) approaches 0.

What is the limit of \( \lim_{x \to 0} \frac{x^{1000} \sin(5x)}{\sin(999x)} \)?

Solve the limit problem \( \lim_{x \to 0} \frac{x^{1000} \sin(5x)}{\sin(999x)} \) using small-angle approximation. Understand the detailed steps and key concepts such as trigonometric limits and small-angle approximations. Suitable for advanced high school or college 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