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
Question
Hi! I just need this. Thank you!
![**Question 3: Determine whether the function is continuous at \( x = 5 \).**
\[ f(x) = \begin{cases}
\frac{x^2 - 4x - 5}{x - 5} & \text{if } x < 5 \\
x + 2 & \text{if } x \geq 5
\end{cases} \]
This problem asks us to evaluate the continuity of the function \( f(x) \) at the point \( x = 5 \). The function \( f(x) \) is defined piecewise with two different expressions depending on the value of \( x \):
1. For \( x < 5 \), \( f(x) \) is given by the rational function \(\frac{x^2 - 4x - 5}{x - 5}\).
2. For \( x \geq 5 \), \( f(x) \) is defined by the linear function \( x + 2 \).
To determine the continuity at \( x = 5 \), we need to check whether the following three conditions are satisfied:
1. \( f(5) \) is defined.
2. The limit of \( f(x) \) as \( x \) approaches 5 from the left ( \( \lim_{x \to 5^{-}} f(x) \) ) exists.
3. The limit of \( f(x) \) as \( x \) approaches 5 from the right ( \( \lim_{x \to 5^{+}} f(x) \) ) exists.
Additionally, the value of \( f(5) \) should match both the left-hand limit and the right-hand limit.
Let's analyze and solve these conditions step-by-step to ensure the function is continuous at \( x = 5 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fef3b440f-f5de-4a2a-817d-48b2b1baa1e0%2F69adf44f-14be-4991-a1d9-8876ada6d57e%2Fiyd9s49_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question 3: Determine whether the function is continuous at \( x = 5 \).**
\[ f(x) = \begin{cases}
\frac{x^2 - 4x - 5}{x - 5} & \text{if } x < 5 \\
x + 2 & \text{if } x \geq 5
\end{cases} \]
This problem asks us to evaluate the continuity of the function \( f(x) \) at the point \( x = 5 \). The function \( f(x) \) is defined piecewise with two different expressions depending on the value of \( x \):
1. For \( x < 5 \), \( f(x) \) is given by the rational function \(\frac{x^2 - 4x - 5}{x - 5}\).
2. For \( x \geq 5 \), \( f(x) \) is defined by the linear function \( x + 2 \).
To determine the continuity at \( x = 5 \), we need to check whether the following three conditions are satisfied:
1. \( f(5) \) is defined.
2. The limit of \( f(x) \) as \( x \) approaches 5 from the left ( \( \lim_{x \to 5^{-}} f(x) \) ) exists.
3. The limit of \( f(x) \) as \( x \) approaches 5 from the right ( \( \lim_{x \to 5^{+}} f(x) \) ) exists.
Additionally, the value of \( f(5) \) should match both the left-hand limit and the right-hand limit.
Let's analyze and solve these conditions step-by-step to ensure the function is continuous at \( x = 5 \).
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 2 steps with 2 images
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