f (x) (A) B = (D x² x- Let f be the function defined above. Which of the following statements about fare true? 1. fhas a limit at x = 2. II. fis continuous at x = 2. III. fis differentiable at x = 2. 1 I only Il only if x # 2 if x = 2 Ill only I and II only II and III
f (x) (A) B = (D x² x- Let f be the function defined above. Which of the following statements about fare true? 1. fhas a limit at x = 2. II. fis continuous at x = 2. III. fis differentiable at x = 2. 1 I only Il only if x # 2 if x = 2 Ill only I and II only II and III
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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![The image displays a calculus problem related to limits, continuity, and differentiability of a piecewise function.
### Problem Statement:
The function \( f(x) \) is defined as follows:
\[
f(x) =
\begin{cases}
\frac{x^2 - 4}{x - 2} & \text{if } x \neq 2 \\
1 & \text{if } x = 2
\end{cases}
\]
You are asked to determine which of the following statements about \( f \) are true:
I. \( f \) has a limit at \( x = 2 \).
II. \( f \) is continuous at \( x = 2 \).
III. \( f \) is differentiable at \( x = 2 \).
### Options:
A) I only
B) II only
C) III only
D) I and II only
E) I, II, and III
To solve this problem, evaluate the limit of \( f(x) \) as \( x \) approaches 2, check the continuity at \( x = 2 \), and determine the differentiability at this point.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0056c178-06b1-466a-850b-6cad9735a7ac%2F4cce314e-dc96-45b5-8496-26614ec3592c%2Fuhnanr7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image displays a calculus problem related to limits, continuity, and differentiability of a piecewise function.
### Problem Statement:
The function \( f(x) \) is defined as follows:
\[
f(x) =
\begin{cases}
\frac{x^2 - 4}{x - 2} & \text{if } x \neq 2 \\
1 & \text{if } x = 2
\end{cases}
\]
You are asked to determine which of the following statements about \( f \) are true:
I. \( f \) has a limit at \( x = 2 \).
II. \( f \) is continuous at \( x = 2 \).
III. \( f \) is differentiable at \( x = 2 \).
### Options:
A) I only
B) II only
C) III only
D) I and II only
E) I, II, and III
To solve this problem, evaluate the limit of \( f(x) \) as \( x \) approaches 2, check the continuity at \( x = 2 \), and determine the differentiability at this point.
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