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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 15c
Chapter 2, Problem 15c

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>
limx→1+f(x)\(\lim\)_{x\(\to\)1^{+}}f\(\left\)(x\(\right\))

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Identify the behavior of the function \( f(x) \) as \( x \) approaches 1 from the right (\( x \to 1^+ \)).
Examine the graph of \( f(x) \) to observe the values of \( f(x) \) as \( x \) gets closer to 1 from values greater than 1.
Determine if \( f(x) \) approaches a specific value, or if it diverges or oscillates as \( x \to 1^+ \).
If \( f(x) \) approaches a specific value, that value is the right-hand limit, \( \lim_{x\to1^{+}}f(x) \).
If \( f(x) \) does not approach a specific value, state that the limit does not exist and provide a reason based on the graph's behavior.

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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 how functions behave near points of interest, including points where they may not be defined. For example, the limit of f(x) as x approaches 1 from the right (denoted as lim x→1⁺ f(x)) examines the values f(x) takes as x gets closer to 1 from values greater than 1.
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One-Sided Limits

One-Sided Limits

One-sided limits are specific types of limits that consider the behavior of a function as the input approaches a particular value from one side only. The right-hand limit (lim x→1⁺ f(x)) looks at values approaching from the right, while the left-hand limit (lim x→1⁻ f(x)) considers values approaching from the left. Understanding one-sided limits is crucial for determining the overall limit at a point, especially when the function exhibits different behaviors from each side.
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One-Sided Limits

Existence of Limits

A limit exists if both the left-hand and right-hand limits at a point are equal. If they differ, the limit does not exist. Additionally, if the function approaches infinity or oscillates without settling at a value, the limit is also considered non-existent. Analyzing the existence of limits is essential for understanding continuity and the overall behavior of functions at specific points.
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Cases Where Limits Do Not Exist
Related Practice
Textbook Question

Suppose g(x) = {2x+1 if x≠0

5 if x=0.


Compute g(0) and lim x→0 g(x)

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Textbook Question

Suppose p and q are polynomials. If lim x→0 p(x) / q(x)=10 and q(0)=2, find p(0).

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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>


a. f(1)

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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>

limx→1−f(x)\(\lim\)_{x\(\to\)1^{-}}f\(\left\)(x\(\right\))

Textbook Question

The position of an object moving vertically along a line is given by the function s(t)=−4.9t2+30t+20s\(\left\)(t\(\right\))=-4.9t^2+30t+20. Find the average velocity of the object over the following intervals.

[0,h]\(\left\[\lbrack\)0,h\(\right\]\rbrack\), where h>0h\(\gt{0}\) is a real number

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Textbook Question

Determine the intervals of continuity for the parking cost function c introduced at the outset of this section (see figure). Consider 0≤t≤60. <FIGURE>

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