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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 2e
Chapter 2, Problem 2e

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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e. limx→1+ f(x) = 1

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To determine the limit of f(x) as x approaches 1 from the right (denoted as limx→1+ f(x)), we need to analyze the behavior of the function as x gets closer to 1 from values greater than 1.
Examine the graph of the function y = f(x) and focus on the section where x is slightly greater than 1. Observe the y-values that the function approaches as x approaches 1 from the right.
Identify the trend of the function's y-values as x approaches 1 from the right. If the y-values are approaching a specific number, that number is the right-hand limit.
Check if the y-value that the function approaches as x approaches 1 from the right is equal to 1. This will help determine if the statement limx→1+ f(x) = 1 is true.
Conclude whether the statement is true or false based on the observed behavior of the function on the graph as x approaches 1 from the right.

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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. For example, the limit of f(x) as x approaches 1 from the right (denoted as lim x→1+ f(x)) examines the value that f(x) approaches 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 limits that consider the behavior of a function as the input approaches a specific point from one side only. The right-hand limit (lim x→c+) looks at values approaching c from the right, while the left-hand limit (lim x→c-) looks at values approaching c from the left. Understanding one-sided limits is crucial for analyzing functions that may have different behaviors on either side of a point.
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One-Sided Limits

Graphical Interpretation of Limits

Graphical interpretation of limits involves analyzing the graph of a function to determine the value that the function approaches as the input approaches a specific point. By observing the graph, one can visually assess whether the limit exists and what value it approaches. This method is particularly useful for identifying discontinuities and understanding the overall behavior of the function near critical points.
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Finding Limits Numerically and Graphically
Related Practice
Textbook Question

Finding Limits


For the function f whose graph is given, determine the following limits. Write ∞ or −∞ where appropriate.


h. lim x → ∞ f(x)


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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?


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g. limx→0+ f(x) = limx→0− f(x)

Textbook Question

Limits and Continuity

On what intervals are the following functions continuous?


d. k(x) = sin x / x

Textbook Question

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


h. f(0)=0


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?

Textbook Question

Limits and Continuity

On what intervals are the following functions continuous?


d. k(x) = x⁻¹/⁶