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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.P.1
Chapter 2, Problem 2.P.1

Limits and Continuity


Graph the function


1 , x ≤ ―1
―x , ―1 < x < 0
ƒ(x) = { 1 , x = 0 ,
―x , 0 < x < 1
1 , x ≥ 1


Then discuss, in detail, limits, one-sided limits, continuity, and one-sided continuity of ƒ at x = ―1 , 0 , and 1. Are any of the discontinuities removable? Explain.

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1
Step 1: Identify the piecewise function and its components. The function ƒ(x) is defined as follows: ƒ(x) = 1 for x ≤ -1, ƒ(x) = -1/x for -1 < x < 0, ƒ(x) = 1 for x = 0, ƒ(x) = -x for 0 < x < 1, and ƒ(x) = 1 for x ≥ 1.
Step 2: Analyze the limits and one-sided limits at x = -1. Evaluate the left-hand limit as x approaches -1 from the left (x → -1⁻) and the right-hand limit as x approaches -1 from the right (x → -1⁺). Compare these limits to the value of the function at x = -1 to determine continuity.
Step 3: Analyze the limits and one-sided limits at x = 0. Evaluate the left-hand limit as x approaches 0 from the left (x → 0⁻) and the right-hand limit as x approaches 0 from the right (x → 0⁺). Compare these limits to the value of the function at x = 0 to determine continuity.
Step 4: Analyze the limits and one-sided limits at x = 1. Evaluate the left-hand limit as x approaches 1 from the left (x → 1⁻) and the right-hand limit as x approaches 1 from the right (x → 1⁺). Compare these limits to the value of the function at x = 1 to determine continuity.
Step 5: Discuss the types of discontinuities at x = -1, 0, and 1. Determine if any discontinuities are removable by checking if redefining the function at these points can make the function continuous. A discontinuity is removable if the limit exists at that point, but the function value is different or undefined.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits describe the behavior of a function as the input approaches a certain value. For the function ƒ(x), understanding limits at x = -1, 0, and 1 involves evaluating the function's value as x approaches these points from both sides. This helps determine if the function approaches a specific value or diverges, which is crucial for analyzing continuity and discontinuities.
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One-Sided Limits

One-Sided Limits

One-sided limits consider the behavior of a function as the input approaches a specific value from one side only, either from the left (x → c⁻) or the right (x → c⁺). For ƒ(x), examining one-sided limits at x = -1, 0, and 1 helps identify potential discontinuities and whether the function behaves differently when approaching these points from different directions.
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One-Sided Limits

Continuity and Discontinuity

A function is continuous at a point if the limit exists and equals the function's value at that point. Discontinuities occur when this condition fails. For ƒ(x), analyzing continuity at x = -1, 0, and 1 involves checking if the limits and function values match. Removable discontinuities can be fixed by redefining the function at the point, while non-removable ones indicate a more fundamental break in the function's behavior.
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Intro to Continuity
Related Practice
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.


f(x)=x³+1


a. [2, 3]

Textbook Question

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limt→0 2t / tan t

Textbook Question

Limits and Continuity

Repeat the instructions of Exercise 1 for


1 , x ≤ ―1

1/x , 0 < |x| < 1

ƒ(x) = { 0, x = 1 ,

1 , x > 1 .

Textbook Question

Centering Intervals About a Point


In Exercises 1–6, sketch the interval (a,b), on the x-axis with the point c inside. Then find a value of δ>0 such that a < x < b whenever 0 < |x−c| < δ.


a=4/9, b=4/7, c=1/2

Textbook Question

Limits and Continuity

On what intervals are the following functions continuous?


a. ƒ(x) = tan x

Textbook Question

Finding Limits Graphically


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

2, x = 2

x/2, x > 2


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a. Find limx→2+ f(x), limx→2− f(x), and f(2).