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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 2.4.4c
Chapter 2, Problem 2.4.4c

Finding Limits Graphically


Let f(x) = {3 - x , x < 2
2, x = 2
x/2, x > 2


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c. Find limx→−1− f(x) and limx→−1+ f(x).

Verified step by step guidance
1
Identify the piecewise function f(x) and the conditions for each piece. Here, f(x) = 3 - x for x < 2 and f(x) = x/2 for x > 2.
Understand that the limit as x approaches -1 from the left, denoted as lim_{x→−1−} f(x), involves considering values of x that are slightly less than -1.
Since -1 is less than 2, use the piece of the function f(x) = 3 - x to evaluate the left-hand limit. Substitute x = -1 into this expression to find the limit.
For the limit as x approaches -1 from the right, denoted as lim_{x→−1+} f(x), consider values of x that are slightly greater than -1.
Again, since -1 is less than 2, use the same piece of the function f(x) = 3 - x to evaluate the right-hand limit. Substitute x = -1 into this expression to find the limit.

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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 of discontinuity. Limits can be approached from the left (denoted as lim x→c−) or from the right (denoted as lim x→c+), which is crucial for analyzing piecewise functions.
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Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In the given function f(x), different formulas apply for x values less than, equal to, or greater than 2. Understanding how to evaluate piecewise functions is essential for finding limits, especially when the limit point lies at a boundary between pieces.
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Graphical Interpretation of Limits

Graphically interpreting limits involves analyzing the behavior of a function's graph as it approaches a specific x-value. This includes observing the values the function approaches from both the left and right sides. For the question at hand, examining the graph of f(x) will help determine the left-hand limit (lim x→−1− f(x)) and the right-hand limit (lim x→−1+ f(x)).
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Related Practice
Textbook Question

Limits and Continuity

On what intervals are the following functions continuous?


c. h(x) = x⁻²/³

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Textbook Question

Horizontal and Vertical Asymptotes


Use limits to determine the equations for all horizontal asymptotes.

_________

/ x² + 9

d. y = / -------------

√ 9x² + 1

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Textbook Question

Finding Limits Graphically


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



d. limx→1− f(x) = 2

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Textbook Question

Limits and Continuity


On what intervals are the following functions continuous?


c. h(x) = cos x / x―π

221
views
Textbook Question

Horizontal and Vertical Asymptotes


Use limits to determine the equations for all horizontal asymptotes.

_____

√x² + 4

c. g(x) = -----------

x

363
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Textbook Question

Finding Limits Graphically


Graph the functions in Exercises 9 and 10. Then answer these questions.


f(x) = {x,−1 ≤ x < 0, or 0 < x ≤ 1

1, x = 0

0, x < −1 or x > 1


d. At what points does the right-hand limit exist but not the left-hand limit?

310
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