Sx² 15 a. Find lim f(x). if x + 2 2. Let f(x) = } if x = 2 X→2 b. Find f(2). Is f(x) continuous at x = 2? с. Explain. d. Sketch a graph of f(x).
Sx² 15 a. Find lim f(x). if x + 2 2. Let f(x) = } if x = 2 X→2 b. Find f(2). Is f(x) continuous at x = 2? с. Explain. d. Sketch a graph of f(x).
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...
Related questions
Concept explainers
Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Question
100%
![Certainly! Below is the transcription and explanation for an educational website:
---
### Problem 2
#### Let \( f(x) = \begin{cases}
x^2 & \text{if } x \neq 2 \\
5 & \text{if } x = 2
\end{cases} \)
**a. Find \( \lim_{{x \to 2}} f(x) \).**
**b. Find \( f(2) \).**
**c. Is \( f(x) \) continuous at \( x = 2 \)? Explain.**
**d. Sketch a graph of \( f(x) \).**
---
#### Explanation:
This problem involves a piecewise function \( f(x) \) which has different definitions depending on the value of \( x \). Specifically:
- For all \( x \) values not equal to 2, the function is defined as \( f(x) = x^2 \).
- For \( x \) equal to 2, the function takes on the value \( f(2) = 5 \).
#### Steps to Solve:
**a. Finding \( \lim_{{x \to 2}} f(x) \):**
To determine the limit of \( f(x) \) as \( x \) approaches 2, we consider the value of the function as \( x \) gets arbitrarily close to 2, but not equal to 2.
Since the function \( f(x) = x^2 \) for all \( x \neq 2 \):
\[ \lim_{{x \to 2}} f(x) = \lim_{{x \to 2}} x^2 = 2^2 = 4 \]
**b. Finding \( f(2) \):**
From the definition of the piecewise function:
\[ f(2) = 5 \]
**c. Is \( f(x) \) continuous at \( x = 2 \)? Explain:**
To determine if \( f(x) \) is continuous at \( x = 2 \), we check the following:
1. The limit of \( f(x) \) as \( x \) approaches 2 exists: \( \lim_{{x \to 2}} f(x) = 4 \)
2. \( f(2) \) is defined: \( f(2) = 5 \)
3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad3ee529-81b1-486a-8459-b4eea2f68af5%2F39cdd3bd-d731-4bb4-9e92-3097e2ad8ddc%2Frk6pn2r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Below is the transcription and explanation for an educational website:
---
### Problem 2
#### Let \( f(x) = \begin{cases}
x^2 & \text{if } x \neq 2 \\
5 & \text{if } x = 2
\end{cases} \)
**a. Find \( \lim_{{x \to 2}} f(x) \).**
**b. Find \( f(2) \).**
**c. Is \( f(x) \) continuous at \( x = 2 \)? Explain.**
**d. Sketch a graph of \( f(x) \).**
---
#### Explanation:
This problem involves a piecewise function \( f(x) \) which has different definitions depending on the value of \( x \). Specifically:
- For all \( x \) values not equal to 2, the function is defined as \( f(x) = x^2 \).
- For \( x \) equal to 2, the function takes on the value \( f(2) = 5 \).
#### Steps to Solve:
**a. Finding \( \lim_{{x \to 2}} f(x) \):**
To determine the limit of \( f(x) \) as \( x \) approaches 2, we consider the value of the function as \( x \) gets arbitrarily close to 2, but not equal to 2.
Since the function \( f(x) = x^2 \) for all \( x \neq 2 \):
\[ \lim_{{x \to 2}} f(x) = \lim_{{x \to 2}} x^2 = 2^2 = 4 \]
**b. Finding \( f(2) \):**
From the definition of the piecewise function:
\[ f(2) = 5 \]
**c. Is \( f(x) \) continuous at \( x = 2 \)? Explain:**
To determine if \( f(x) \) is continuous at \( x = 2 \), we check the following:
1. The limit of \( f(x) \) as \( x \) approaches 2 exists: \( \lim_{{x \to 2}} f(x) = 4 \)
2. \( f(2) \) is defined: \( f(2) = 5 \)
3.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning