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y=fx) 3 2+ 2 3 45 6 a=3 lim f(x) = lim f(x) = lim f(x)

y=fx) 3 2+ 2 3 45 6 a=3 lim f(x) = lim f(x) = lim f(x)

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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### 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)
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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