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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2b
Chapter 2, Problem 2b

Finding Limits Graphically


Which of the following statements about the function y = f(x) graphed here are true, and which are false?


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b. limx→2 f(x) does not exist

Verified step by step guidance
1
Examine the graph of the function y = f(x) around x = 2 to determine the behavior of the function as x approaches 2 from both the left and the right.
Identify the left-hand limit, which is the value that f(x) approaches as x approaches 2 from the left (x → 2⁻).
Identify the right-hand limit, which is the value that f(x) approaches as x approaches 2 from the right (x → 2⁺).
Compare the left-hand limit and the right-hand limit. For the overall limit lim(x→2) f(x) to exist, these two limits must be equal.
If the left-hand limit and the right-hand limit are not equal, then the limit lim(x→2) f(x) does not exist. Otherwise, if they are equal, the limit exists and is equal to that common value.

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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 determine the value that a function approaches, even if it does not actually reach that value. Understanding limits is crucial for analyzing continuity, derivatives, and integrals.
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One-Sided Limits

Graphical Interpretation of Limits

Graphically, the limit of a function as x approaches a certain value can be observed by examining the behavior of the function's graph near that point. If the function approaches a specific y-value from both sides as x approaches a given value, the limit exists. However, if the function behaves differently from the left and right, the limit may not exist.
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Existence of Limits

For a limit to exist at a point, the left-hand limit and right-hand limit must both exist and be equal. If there is a discontinuity, such as a jump or an asymptote at that point, the limit does not exist. This concept is essential for determining the validity of statements regarding limits in a given function.
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Related Practice
Textbook Question

Use the graph of the greatest integer function y = ⌊x⌋, Figure 1.10 in Section 1.1, to help you find the limits in Exercises 21 and 22.


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b. limt→4−(t−⌊t⌋)

Textbook Question

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.


lim ( 1 / x¹/³ − 1 / (x − 1)⁴/³ ) as


a. x → 0⁺

Textbook Question

Average Rates of Change


In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.


g(t)=2+cos t


b. [0,π]

Textbook Question

Limits and Continuity

On what intervals are the following functions continuous?


b. g(x) = x³/⁴

Textbook Question

Estimating Limits


[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.


Let f(x) = (x² - 9) / (x + 3)


b. Support your conclusions in part (a) by graphing f near c = -3 and using Zoom and Trace to estimate y-values on the graph as x → −3.

Textbook Question

Estimating Limits


[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.


Let G(x)=(x + 6)/(x² + 4x − 12)


b. Support your conclusions in part (a) by graphing G and using Zoom and Trace to estimate y-values on the graph as x→−6.