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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 2 - LimitsProblema 18i
Capitolo 2, Problema 18i

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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1
Identify the behavior of the function g(x) as x approaches 4 from the left (x -> 4^-).
Identify the behavior of the function g(x) as x approaches 4 from the right (x -> 4^+).
Compare the left-hand limit and the right-hand limit of g(x) as x approaches 4.
If both the left-hand limit and the right-hand limit are equal, then the limit \( \lim_{x\to4}g(x) \) exists and is equal to that common value.
If the left-hand limit and the right-hand limit are not equal, then the limit \( \lim_{x\to4}g(x) \) does not exist.

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Limits

A limit describes the behavior of a function as its input approaches a certain value. It is essential for understanding continuity and the behavior of functions at specific points. In this context, we are interested in the limit of g(x) as x approaches 4, which can indicate whether g(x) approaches a specific value or diverges.
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One-Sided Limits

Continuity

A function is continuous at a point if the limit as x approaches that point equals the function's value at that point. For the limit of g(x) as x approaches 4 to exist, g(4) must also be defined and equal to the limit. Discontinuities can arise from jumps, holes, or vertical asymptotes in the graph.
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Intro to Continuity

Existence of Limits

The existence of a limit requires that the left-hand limit and right-hand limit at a point are equal. If they differ or if one of them does not exist, then the overall limit does not exist. In analyzing g(x) at x = 4, one must check the behavior of the function from both sides of 4 to determine if the limit exists.
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Cases Where Limits Do Not Exist
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