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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![### Continuity Check of Function \( f(x) \)
Determine if \( f \) is continuous at the indicated values.
The function \( f(x) \) is defined as follows:
\[
f(x) =
\begin{cases}
1 & \text{if } x = 0 \\
\frac{\sin x}{x} & \text{if } x \neq 0
\end{cases}
\]
**Questions to Determine Continuity:**
a) Is \( f \) continuous at \( x = 0 \)?
- Select your response from the dropdown menu.
b) Is \( f \) continuous at \( x = -3\pi \)?
- Select your response from the dropdown menu.
**Explanation:**
To determine the continuity of \( f \) at a point \( x = c \):
1. \( f(c) \) must be defined.
2. The limit of \( f(x) \) as \( x \) approaches \( c \) from both sides must exist.
3. The limit of \( f(x) \) as \( x \) approaches \( c \) must be equal to \( f(c) \).
Evaluate these conditions for:
- \( x = 0 \)
- \( x = -3\pi \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa043fe4-4451-42e8-82d1-21cc59f9ce63%2Fbc3e2f42-525a-4385-92cb-94836c3a51e5%2F6nfvxcj.png&w=3840&q=75)
Transcribed Image Text:### Continuity Check of Function \( f(x) \)
Determine if \( f \) is continuous at the indicated values.
The function \( f(x) \) is defined as follows:
\[
f(x) =
\begin{cases}
1 & \text{if } x = 0 \\
\frac{\sin x}{x} & \text{if } x \neq 0
\end{cases}
\]
**Questions to Determine Continuity:**
a) Is \( f \) continuous at \( x = 0 \)?
- Select your response from the dropdown menu.
b) Is \( f \) continuous at \( x = -3\pi \)?
- Select your response from the dropdown menu.
**Explanation:**
To determine the continuity of \( f \) at a point \( x = c \):
1. \( f(c) \) must be defined.
2. The limit of \( f(x) \) as \( x \) approaches \( c \) from both sides must exist.
3. The limit of \( f(x) \) as \( x \) approaches \( c \) must be equal to \( f(c) \).
Evaluate these conditions for:
- \( x = 0 \)
- \( x = -3\pi \)
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