what is the limit as x approaches 0 when f(x)= sin(5x)/x

The limit of \( f(x) = \frac{\sin(5x)}{x} \) as \( x \) approaches 0 can be evaluated using standard trigonometric limit properties.

We know that:

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

Let's manipulate the given function \( \frac{\sin(5x)}{x} \). We can rewrite it as:

\[
\frac{\sin(5x)}{x} = \frac{\sin(5x)}{5x} \cdot 5
\]

Now, using the fact that:

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

We multiply by 5:

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

Thus, the limit is:

\[
\boxed{5}
\]

Would you like more details on this process or have any other questions?

Here are 5 related questions:
1. What is the limit of \( \frac{\sin(kx)}{x} \) as \( x \to 0 \) for any constant \( k \)?
2. Can you use L'Hopital's Rule to solve this limit? How?
3. How does the small-angle approximation for sine function help in evaluating limits?
4. What would the limit be if we had \( \frac{\sin(5x)}{x^2} \) as \( x \to 0 \)?
5. What happens to the limit when you replace sine with another function like cosine?

**Tip:** Whenever dealing with trigonometric limits, especially involving sine and cosine, the small-angle approximation is very useful for simplifying expressions.

What is the limit as x approaches 0 when f(x) = sin(5x) / x?

Learn how to find the limit of sin(5x)/x as x approaches 0 with this step-by-step guide. This problem involves using the small-angle approximation for trigonometric functions and standard limit properties. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation