https://r2.erweima.ai/mathbot/image/1726973119879bj2uvigt.jpg

The problem you've provided asks for the determination of the value of \( a \) so that the function \( f(x) \) is continuous over \( \mathbb{R} \). The function is given as:

\[
f(x) = \begin{cases} 
\frac{\sin(2x)}{3x}, & x \neq 0 \\
a, & x = 0 
\end{cases}
\]

To ensure continuity at \( x = 0 \), the limit of \( f(x) \) as \( x \) approaches 0 must equal \( a \). We find this limit:

\[
\lim_{x \to 0} \frac{\sin(2x)}{3x} 
\]

Using the small angle approximation for sine (\(\sin x \approx x\) when \( x \) is small), we have:

\[
\sin(2x) \approx 2x
\]

Thus,

\[
\lim_{x \to 0} \frac{\sin(2x)}{3x} = \lim_{x \to 0} \frac{2x}{3x} = \frac{2}{3}
\]

Therefore, \( a \) must be \( \frac{2}{3} \) to make \( f(x) \) continuous over all real numbers.

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

Here are five related questions you could consider:
1. What would change if the numerator of the fraction was \(\sin(3x)\) instead of \(\sin(2x)\)?
2. How would the function behave near \( x = 0 \) if \( a \) were set to a different value?
3. Can you explain the "small angle approximation" used to simplify the sine function?
4. What are the implications of this function's behavior for integration around \( x = 0 \)?
5. What would be the derivative of \( f(x) \) at \( x = 0 \) if \( a \) is correctly chosen as \( \frac{2}{3} \)?

**Tip:** When dealing with limits involving trigonometric functions, using L'Hôpital's Rule can also provide a rigorous way to determine limits if the direct substitution leads to an indeterminate form like \( \frac{0}{0} \).

Cho hàm số f(x) = { sin(2x)/3x, x ≠ 0; a, x = 0. Xác định giá trị của a để f liên tục trên R.

Learn how to determine the value of 'a' for which a given piecewise function, involving the trigonometric expression sin(2x)/3x, is continuous over R. Explore key concepts like limits, small angle approximation, and continuity. Suitable for high school students 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