Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Problem Statement:**
Find the constant \( a \) such that the function is continuous on the entire real line.
**Function Definition:**
\[
g(x) =
\begin{cases}
\frac{4\sin x}{x} & \text{if } x < 0 \\
a - 4x & \text{if } x \geq 0
\end{cases}
\]
**Solution Input:**
\[ a = \boxed{} \]
**Explanation:**
The problem gives a piecewise function, \( g(x) \), and asks for the determination of the constant \( a \) that ensures \( g(x) \) is continuous across all real numbers. The function has two expressions, one for \( x < 0 \) and another for \( x \geq 0 \). Continuity across the real line means these two parts of the function must meet smoothly at \( x = 0 \). To find this \( a \), we need to ensure the limit of \( \frac{4\sin x}{x} \) as \( x \) approaches 0 equals the value of \( a - 4(0) \), which is \( a \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9ba58da1-aa6c-4b34-ac65-53856b97f0bf%2F5ec4c2a7-2626-4015-8c8b-2f85524c263b%2Fxbvbrk6_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the constant \( a \) such that the function is continuous on the entire real line.
**Function Definition:**
\[
g(x) =
\begin{cases}
\frac{4\sin x}{x} & \text{if } x < 0 \\
a - 4x & \text{if } x \geq 0
\end{cases}
\]
**Solution Input:**
\[ a = \boxed{} \]
**Explanation:**
The problem gives a piecewise function, \( g(x) \), and asks for the determination of the constant \( a \) that ensures \( g(x) \) is continuous across all real numbers. The function has two expressions, one for \( x < 0 \) and another for \( x \geq 0 \). Continuity across the real line means these two parts of the function must meet smoothly at \( x = 0 \). To find this \( a \), we need to ensure the limit of \( \frac{4\sin x}{x} \) as \( x \) approaches 0 equals the value of \( a - 4(0) \), which is \( a \).
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