The limit you're asking for is:

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

To solve this, you can manipulate the expression using the standard trigonometric limit \(\lim_{x \to 0} \frac{\sin(x)}{x} = 1\):

\[
\lim_{x \to 0} \frac{\sin(5x)}{x} = \lim_{x \to 0} \left( 5 \cdot \frac{\sin(5x)}{5x} \right)
\]

Since \(5x\) approaches 0 as \(x\) approaches 0, we can use the standard limit:

\[
\lim_{x \to 0} \frac{\sin(5x)}{5x} = 1
\]

So, the expression simplifies to:

\[
\lim_{x \to 0} 5 \cdot 1 = 5
\]

Thus, the limit is \(5\).

Would you like further clarification, or do you have any questions?

### 5 Related Questions:
1. Why does \(\lim_{x \to 0} \frac{\sin(x)}{x} = 1\)?
2. How can trigonometric identities help in solving limits?
3. What is the significance of using substitution in solving limits?
4. How would you solve \(\lim_{x \to 0} \frac{\tan(x)}{x}\)?
5. What happens to \(\frac{\sin(5x)}{x}\) as \(x\) approaches infinity?

### Tip:
When working with limits involving trigonometric functions, especially as the variable approaches 0, substituting the angle with a simple multiple of \(x\) can simplify the process by leveraging known limits like \(\frac{\sin(x)}{x}\).

Explore the solution to the limit problem lim_{x \to 0} \frac{\sin(5x)}{x}. Understand how trigonometric identities and limits work together to find the result. Suitable for advanced high school students and anyone interested in calculus and trigonometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation