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3. Sketch the graph of a function f such that the following conditions are true: ● f(4) is not defined, f(1) is defined, but limx→1 f(x) does not exist. f(-2) is defined and limx→-2 f(x) exists, but limx→-2 f(x) ‡ f(−2) ● f is continuous for all other x-values

3. Sketch the graph of a function f such that the following conditions are true: ● f(4) is not defined, f(1) is defined, but limx→1 f(x) does not exist. f(-2) is defined and limx→-2 f(x) exists, but limx→-2 f(x) ‡ f(−2) ● f is continuous for all other x-values

Calculus: Early Transcendentals
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 Statement 3. Sketch the graph of a function \( f \) such that the following conditions are true: - \( f(4) \) is not defined, - \( f(1) \) is defined, but \( \lim_{x \to 1} f(x) \) does not exist, - \( f(-2) \) is defined and \( \lim_{x \to -2} f(x) \) exists, but \( \lim_{x \to -2} f(x) \neq f(-2) \), - \( f \) is continuous for all other \( x \)-values ## Graph Description The graph provided is a standard Cartesian coordinate plane with the x-axis and y-axis intersecting at the origin \((0, 0)\). This forms four quadrants which can be used to sketch the function \( f \) that meets the specified conditions. ### Conditions Explained 1. **\( f(4) \) is not defined:** - At \( x = 4 \), there should be a point missing on the graph to indicate that the function value is not defined there. 2. **\( f(1) \) is defined, but \( \lim_{x \to 1} f(x) \) does not exist:** - At \( x = 1 \), the function \( f \) has a definite value, but there should be a discontinuity around \( x = 1 \), perhaps from a jump in the function values as \( x \) approaches 1 from the left and right. 3. **\( f(-2) \) is defined and \( \lim_{x \to -2} f(x) \) exists, but \( \lim_{x \to -2} f(x) \neq f(-2) \):** - At \( x = -2 \), the function \( f \) should converge to a different value than \( f(-2) \), which could be shown by a filled dot at \( ( -2 , f(-2) ) \) and a hollow dot at another value along \( x = -2 \). 4. **\( f \) is continuous for all other \( x \)-values:** - Apart from the points where the above conditions hold, the function \( f \) should be continuous elsewhere on the graph. Use these conditions to appropriately sketch
Transcribed Image Text:## Problem Statement 3. Sketch the graph of a function \( f \) such that the following conditions are true: - \( f(4) \) is not defined, - \( f(1) \) is defined, but \( \lim_{x \to 1} f(x) \) does not exist, - \( f(-2) \) is defined and \( \lim_{x \to -2} f(x) \) exists, but \( \lim_{x \to -2} f(x) \neq f(-2) \), - \( f \) is continuous for all other \( x \)-values ## Graph Description The graph provided is a standard Cartesian coordinate plane with the x-axis and y-axis intersecting at the origin \((0, 0)\). This forms four quadrants which can be used to sketch the function \( f \) that meets the specified conditions. ### Conditions Explained 1. **\( f(4) \) is not defined:** - At \( x = 4 \), there should be a point missing on the graph to indicate that the function value is not defined there. 2. **\( f(1) \) is defined, but \( \lim_{x \to 1} f(x) \) does not exist:** - At \( x = 1 \), the function \( f \) has a definite value, but there should be a discontinuity around \( x = 1 \), perhaps from a jump in the function values as \( x \) approaches 1 from the left and right. 3. **\( f(-2) \) is defined and \( \lim_{x \to -2} f(x) \) exists, but \( \lim_{x \to -2} f(x) \neq f(-2) \):** - At \( x = -2 \), the function \( f \) should converge to a different value than \( f(-2) \), which could be shown by a filled dot at \( ( -2 , f(-2) ) \) and a hollow dot at another value along \( x = -2 \). 4. **\( f \) is continuous for all other \( x \)-values:** - Apart from the points where the above conditions hold, the function \( f \) should be continuous elsewhere on the graph. Use these conditions to appropriately sketch
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