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 5: Consider the Function
Given the function:
\[
f(x) =
\begin{cases}
e^{-x}, & \text{if } x < 0 \\
\cos x, & \text{if } x \geq 0
\end{cases}
\]
#### (a) Evaluate Limits and Continuity at \(x = 0\)
- Find the left-hand limit as \(x\) approaches 0: \(\lim_{x \to 0^-} f(x)\)
- Find the right-hand limit as \(x\) approaches 0: \(\lim_{x \to 0^+} f(x)\)
- Evaluate the function at \(x = 0\): \(f(0)\)
- Determine whether \(f\) is continuous at \(x = 0\), providing your reasoning.
#### (b) Sketch the Graph of the Function \(f(x)\)
- Visualize and draw the piecewise function based on the given conditions.
### Solution:
#### (a) Evaluating Limits and Continuity:
**Left-hand limit:**
\[
\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} e^{-x} = e^0 = 1
\]
**Right-hand limit:**
\[
\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \cos x = \cos 0 = 1
\]
**Function value at \(x = 0\):**
\[
f(0) = \cos 0 = 1
\]
**Continuity at \(x = 0\):**
A function is continuous at \(x = 0\) if the following three conditions are met:
1. \(\lim_{x \to 0^-} f(x)\) exists.
2. \(\lim_{x \to 0^+} f(x)\) exists.
3. \(\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0)\).
Since \(\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0) = 1\), the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6484a2ba-7b6c-4144-bd9b-1f2d784a131b%2F86a53f9b-a6af-45e9-b908-390d5bfa97c9%2F96oodr8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 5: Consider the Function
Given the function:
\[
f(x) =
\begin{cases}
e^{-x}, & \text{if } x < 0 \\
\cos x, & \text{if } x \geq 0
\end{cases}
\]
#### (a) Evaluate Limits and Continuity at \(x = 0\)
- Find the left-hand limit as \(x\) approaches 0: \(\lim_{x \to 0^-} f(x)\)
- Find the right-hand limit as \(x\) approaches 0: \(\lim_{x \to 0^+} f(x)\)
- Evaluate the function at \(x = 0\): \(f(0)\)
- Determine whether \(f\) is continuous at \(x = 0\), providing your reasoning.
#### (b) Sketch the Graph of the Function \(f(x)\)
- Visualize and draw the piecewise function based on the given conditions.
### Solution:
#### (a) Evaluating Limits and Continuity:
**Left-hand limit:**
\[
\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} e^{-x} = e^0 = 1
\]
**Right-hand limit:**
\[
\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \cos x = \cos 0 = 1
\]
**Function value at \(x = 0\):**
\[
f(0) = \cos 0 = 1
\]
**Continuity at \(x = 0\):**
A function is continuous at \(x = 0\) if the following three conditions are met:
1. \(\lim_{x \to 0^-} f(x)\) exists.
2. \(\lim_{x \to 0^+} f(x)\) exists.
3. \(\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0)\).
Since \(\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0) = 1\), the
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