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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 3d
Chapter 2, Problem 3d

Use the graph of hh in the figure to find the following values or state that they do not exist. <IMAGE>
limx→4h(x){\(\displaystyle\]\lim\)_{x\(\to\)4}h\(\left\)(x\(\right\))}

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1
Identify the point of interest on the graph, which is x = 4, and observe the behavior of the function h(x) as x approaches 4 from both the left and the right.
Examine the left-hand limit, which is the value that h(x) approaches as x approaches 4 from values less than 4. Look at the graph to see where the function is heading as x gets closer to 4 from the left side.
Examine the right-hand limit, which is the value that h(x) approaches as x approaches 4 from values greater than 4. Look at the graph to see where the function is heading as x gets closer to 4 from the right side.
Determine if the left-hand limit and the right-hand limit are equal. If they are equal, then the limit exists and is equal to this common value. If they are not equal, then the limit does not exist.
State the conclusion based on the observations: if the left-hand and right-hand limits are equal, provide that value as the limit. If they are not equal, state that the limit does not exist.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding the function's value at points where it may not be explicitly defined. For example, the limit of h(x) as x approaches 4 indicates what value h(x) approaches as x gets closer to 4, which is crucial for analyzing continuity and differentiability.
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One-Sided Limits

Continuity

Continuity of a function at a point means that the function is defined at that point, the limit exists, and the limit equals the function's value at that point. If h(x) is continuous at x = 4, then the limit as x approaches 4 will equal h(4). Understanding continuity is essential for determining the existence of limits and for applying theorems related to calculus.
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Intro to Continuity

Graph Interpretation

Interpreting graphs is a key skill in calculus that involves analyzing the visual representation of functions to extract information about their behavior. By examining the graph of h(x), one can identify limits, continuity, and points of discontinuity. This visual approach aids in understanding complex concepts and provides insight into the function's overall behavior near specific points.
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Graphing The Derivative
Related Practice
Textbook Question

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→−1^− f(x)

Textbook Question

Use the graph of hh in the figure to find the following values or state that they do not exist. <IMAGE>

h(2)h\(\left\)(2\(\right\))

Textbook Question

What does it mean for a function to be continuous on an interval?

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Textbook Question

Use the graph of hh in the figure to find the following values or state that they do not exist. <IMAGE>

h(4)h\(\left\)(4\(\right\))

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Textbook Question

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

f(−1)

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Textbook Question

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→−1^+ f(x)