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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 18d
Chapter 2, Problem 18d

Use the graph of g(x)g(x) in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>


limx→2g(x)\(\lim\)_{x\(\to\)2}g\(\left\)(x\(\right\))

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1
Examine the graph of g(x) around x = 2 to understand the behavior of the function as x approaches 2 from both the left and the right.
Identify the left-hand limit, \( \lim_{x \to 2^-} g(x) \), by observing the values that g(x) approaches as x approaches 2 from the left side.
Identify the right-hand limit, \( \lim_{x \to 2^+} g(x) \), by observing the values that g(x) approaches as x approaches 2 from the right side.
Compare the left-hand limit and the right-hand limit. If both limits are equal, then the limit \( \lim_{x \to 2} g(x) \) exists and is equal to this common value.
If the left-hand limit and the right-hand limit are not equal, then the limit \( \lim_{x \to 2} g(x) \) does not exist, and you should explain that the discrepancy between the two limits is the reason for the non-existence.

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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 g(x) as x approaches 2 examines how g(x) behaves near x = 2, which is crucial for determining 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 g(x) is continuous at x = 2, then the limit as x approaches 2 will equal g(2). Discontinuities can arise from jumps, holes, or vertical asymptotes, affecting the existence of limits.
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Intro to Continuity

Graphical Interpretation

Graphical interpretation involves analyzing the visual representation of a function to understand its behavior. By examining the graph of g(x), one can identify limits, continuity, and points of discontinuity. This visual approach aids in determining whether the limit as x approaches 2 exists and provides insight into the function's overall behavior near that point.
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Determining Differentiability Graphically
Related Practice
Textbook Question

Determine the following limits. 

lim x→−∞ (5 + 100/x + sin4 x3 / x2)

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Textbook Question

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>


l. limx→2f(x)\(\lim\)_{x\(\to\)2}f\(\left\)(x\(\right\))

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Textbook Question

Use the graph of g(x) in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

limx→4g(x)\(\lim\)_{x\(\to\)4}g\(\left\)(x\(\right\))

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Textbook Question

Determine the following limits. 

lim t→∞ (5t2 + t sin t) / t2

Textbook Question

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.


lim x→4(3x−7)

Textbook Question

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>


h. limx→3f(x)\(\lim\)_{x\(\to\)3}f\(\left\)(x\(\right\))

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