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(a)f(1), (6) lim→1 f(x), (c)f (2), (d) lim,→2, (e)f(3), (f) lim,→3 yA y = f(x) 4 3. -1 1 2 3 4

(a)f(1), (6) lim→1 f(x), (c)f (2), (d) lim,→2, (e)f(3), (f) lim,→3 yA y = f(x) 4 3. -1 1 2 3 4

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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How do I evaluate the limits of 1c, 1d, 1f? 

**Transcription for Educational Website** --- ## Understanding Piecewise Functions and Limits: An Analysis of \(y = f(x)\) ### Given Function and Key Points: The provided graph represents the function \( y = f(x) \). To understand the behavior of the function, focus on specific points and limits as marked by black and white dots on the graph. ### Graph Analysis: 1. **Axes and Grid:** - The \( x \)-axis ranges from \(-1\) to 5. - The \( y \)-axis ranges from 1 to 5. - Major grid lines at integer intervals provide a clear view of function values at those points. 2. **Key Points on the Graph:** - Open circles (○) represent points where the value is not included in the function at that specific \( x \)-value. - Solid circles (•) represent points where the value is included in the function at that specific \( x \)-value. ### Specific Queries: Here are the specific questions regarding the function \( y = f(x) \): 1. **(a) \( f(1) \):** - From the graph, at \( x = 1 \), there is an open circle at \( y = 2 \). This indicates \( f(1) \) does not exist at \( y = 2 \). Instead, look for a solid circle to define \( f(1) \). Hence, \( f(1) = 3 \). 2. **(b) \( \lim_{x \to 1} f(x) \):** - The limit as \( x \) approaches 1 from both sides should meet at a specific y-value. From the function behavior, as \( x \to 1 \), \( f(x) \to 2 \). Therefore, \( \lim_{x \to 1} f(x) = 2 \). 3. **(c) \( f(2) \):** - Directly from the graph, \( f(2) = 4 \). 4. **(d) \( \lim_{x \to 2} \):** - Observe the function’s approach as \( x \) nears 2. The function values approach the y-value at \( x = 2 \), hence \( \lim_{x \to 2} f(x) = 4 \
Transcribed Image Text:**Transcription for Educational Website** --- ## Understanding Piecewise Functions and Limits: An Analysis of \(y = f(x)\) ### Given Function and Key Points: The provided graph represents the function \( y = f(x) \). To understand the behavior of the function, focus on specific points and limits as marked by black and white dots on the graph. ### Graph Analysis: 1. **Axes and Grid:** - The \( x \)-axis ranges from \(-1\) to 5. - The \( y \)-axis ranges from 1 to 5. - Major grid lines at integer intervals provide a clear view of function values at those points. 2. **Key Points on the Graph:** - Open circles (○) represent points where the value is not included in the function at that specific \( x \)-value. - Solid circles (•) represent points where the value is included in the function at that specific \( x \)-value. ### Specific Queries: Here are the specific questions regarding the function \( y = f(x) \): 1. **(a) \( f(1) \):** - From the graph, at \( x = 1 \), there is an open circle at \( y = 2 \). This indicates \( f(1) \) does not exist at \( y = 2 \). Instead, look for a solid circle to define \( f(1) \). Hence, \( f(1) = 3 \). 2. **(b) \( \lim_{x \to 1} f(x) \):** - The limit as \( x \) approaches 1 from both sides should meet at a specific y-value. From the function behavior, as \( x \to 1 \), \( f(x) \to 2 \). Therefore, \( \lim_{x \to 1} f(x) = 2 \). 3. **(c) \( f(2) \):** - Directly from the graph, \( f(2) = 4 \). 4. **(d) \( \lim_{x \to 2} \):** - Observe the function’s approach as \( x \) nears 2. The function values approach the y-value at \( x = 2 \), hence \( \lim_{x \to 2} f(x) = 4 \
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