https://file.aiquickdraw.com/mathbot/file/172613278130954b9w9hj.jpg

To solve this problem, we need to ensure that the given piecewise function \( f(x) \) is continuous on the entire real line, especially at \( x = 0 \), where the two pieces of the function meet.

### Step 1: Analyze the pieces of the function

The function is defined as:

\[
f(x) = \begin{cases}
-\frac{4 \sin x}{x}, & x < 0 \\
a + 7x, & x \geq 0
\end{cases}
\]

For the function to be continuous at \( x = 0 \), we require the left-hand limit (as \( x \to 0^- \)) to equal the right-hand limit (as \( x \to 0^+ \)) and the value of the function at \( x = 0 \).

### Step 2: Find the left-hand limit as \( x \to 0^- \)

As \( x \to 0^- \), the left-hand side of the function is:

\[
\lim_{x \to 0^-} -\frac{4 \sin x}{x}
\]

Using the known limit identity:

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

we get:

\[
\lim_{x \to 0^-} -\frac{4 \sin x}{x} = -4.
\]

### Step 3: Find the right-hand limit as \( x \to 0^+ \)

For \( x \geq 0 \), the function is \( f(x) = a + 7x \). The right-hand limit as \( x \to 0^+ \) is:

\[
\lim_{x \to 0^+} (a + 7x) = a.
\]

### Step 4: Set the limits equal

To ensure continuity, the left-hand limit must equal the right-hand limit at \( x = 0 \):

\[
-4 = a.
\]

Thus, the value of \( a \) is \( -4 \).

### Final Answer:
The correct answer is \( \boxed{-4} \) (Option E).

Would you like a detailed explanation of any step or have any further questions?

Here are 5 related questions to expand your understanding:

1. How do you define the continuity of a function at a point?
2. What is the significance of using the limit identity \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \) in this problem?
3. How do you determine if a piecewise function is continuous at a junction point?
4. How does the right-hand and left-hand limit approach help in solving continuity problems?
5. Can you have a continuous function that is not differentiable at some points?

**Tip:** When checking continuity for piecewise functions, always compare the left-hand and right-hand limits at the point where the pieces meet.

Find the constant a such that the function f(x) is continuous on the entire real line.

Solve the problem of finding the constant 'a' to ensure the continuity of a piecewise function on the entire real line. The problem focuses on key concepts such as limits and continuity for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation