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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![### Analyzing Limits from a Graph
#### Objective:
Use the graph of the function \( f \) to find the limits at the indicated value of \( a \), if the limit exists. (If an answer does not exist, enter DNE.)
#### Graph Description:
The given graph depicts a function \( y = f(x) \) with respect to the \( x \)-axis and \( y \)-axis. The point of interest, where \( a = 3 \), is explicitly marked on the \( x \)-axis. The curve of \( f(x) \) illustrates the following behavior around \( x = 3 \):
- As \( x \) approaches 3 from the left (\( x \to 3^{-} \)), the function \( y \) values are trending towards 4.
- As \( x \) approaches 3 from the right (\( x \to 3^{+} \)), the function \( y \) values are trending towards 2.
- The function \( f(x) \) has an obvious discontinuity at \( x = 3 \), suggesting a potential jump discontinuity.
#### Limit Analysis:
Using the graph, we can assess the following limits:
1. \[\lim_{x \to 3^{-}} f(x)\]:
- As \( x \to 3^{-} \) (from the left), the value of \( f(x) \) tends to 4.
- Therefore, \(\lim_{x \to 3^{-}} f(x) = 4\).
2. \[\lim_{x \to 3^{+}} f(x)\]:
- As \( x \to 3^{+} \) (from the right), the value of \( f(x) \) tends to 2.
- Therefore, \(\lim_{x \to 3^{+}} f(x) = 2\).
3. \[\lim_{x \to 3} f(x)\]:
- Since the left-hand limit (\(\lim_{x \to 3^{-}}\)) and right-hand limit (\(\lim_{x \to 3^{+}}\)) do not equal each other (4 \(\neq\) 2), the general limit \(\lim_{x \to 3} f(x)\) does not exist.
- Therefore, \(\lim_{x \to 3} f(x)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad356ace-94cd-4e8a-aa7f-bdf17308114e%2F33ab980b-92cd-4f94-829e-073e902e7c71%2Fkfw450k.png&w=3840&q=75)
Transcribed Image Text:### Analyzing Limits from a Graph
#### Objective:
Use the graph of the function \( f \) to find the limits at the indicated value of \( a \), if the limit exists. (If an answer does not exist, enter DNE.)
#### Graph Description:
The given graph depicts a function \( y = f(x) \) with respect to the \( x \)-axis and \( y \)-axis. The point of interest, where \( a = 3 \), is explicitly marked on the \( x \)-axis. The curve of \( f(x) \) illustrates the following behavior around \( x = 3 \):
- As \( x \) approaches 3 from the left (\( x \to 3^{-} \)), the function \( y \) values are trending towards 4.
- As \( x \) approaches 3 from the right (\( x \to 3^{+} \)), the function \( y \) values are trending towards 2.
- The function \( f(x) \) has an obvious discontinuity at \( x = 3 \), suggesting a potential jump discontinuity.
#### Limit Analysis:
Using the graph, we can assess the following limits:
1. \[\lim_{x \to 3^{-}} f(x)\]:
- As \( x \to 3^{-} \) (from the left), the value of \( f(x) \) tends to 4.
- Therefore, \(\lim_{x \to 3^{-}} f(x) = 4\).
2. \[\lim_{x \to 3^{+}} f(x)\]:
- As \( x \to 3^{+} \) (from the right), the value of \( f(x) \) tends to 2.
- Therefore, \(\lim_{x \to 3^{+}} f(x) = 2\).
3. \[\lim_{x \to 3} f(x)\]:
- Since the left-hand limit (\(\lim_{x \to 3^{-}}\)) and right-hand limit (\(\lim_{x \to 3^{+}}\)) do not equal each other (4 \(\neq\) 2), the general limit \(\lim_{x \to 3} f(x)\) does not exist.
- Therefore, \(\lim_{x \to 3} f(x)
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