Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

1. Limits and Continuity

Continuity

Calculus

1. Limits and Continuity

Continuity

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4:31 minutes

Problem 2.P.2

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 .

Verified step by step guidance

First, identify the piecewise function given: \( f(x) = \begin{cases} 1, & x \leq -1 \\ 1/x, & 0 < |x| < 1 \\ 0, & x = 1 \\ 1, & x > 1 \end{cases} \).

To analyze the limits and continuity, consider each interval separately. Start with \( x \leq -1 \), where \( f(x) = 1 \). The function is constant, so it is continuous in this interval.

Next, examine the interval \( 0 < |x| < 1 \), where \( f(x) = 1/x \). Check the limit as \( x \) approaches 0 from both sides. Since \( 1/x \) becomes unbounded as \( x \) approaches 0, the function is not continuous at \( x = 0 \).

Consider \( x = 1 \), where \( f(x) = 0 \). Evaluate the limit from the left and right of \( x = 1 \). From the left, \( f(x) = 1/x \) approaches 1, and from the right, \( f(x) = 1 \). Since the limits from both sides do not equal \( f(1) = 0 \), the function is not continuous at \( x = 1 \).

Finally, for \( x > 1 \), \( f(x) = 1 \). The function is constant, so it is continuous in this interval. Summarize the continuity: \( f(x) \) is continuous for \( x \leq -1 \) and \( x > 1 \), but not at \( x = 0 \) or \( x = 1 \).

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Key Concepts

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

Limits

Limits are fundamental in calculus, describing the behavior of a function as its input approaches a particular point. Understanding limits is crucial for analyzing the function's behavior near points of interest, especially where the function may not be explicitly defined. In this context, limits help determine the value that f(x) approaches as x approaches specific values, such as -1, 0, or 1.

Continuity

Continuity of a function at a point means that the function is defined at that point, the limit exists at that point, and the limit equals the function's value. A function is continuous over an interval if it is continuous at every point within that interval. For the given piecewise function, assessing continuity involves checking these conditions at the transition points x = -1, 0, and 1.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain. Understanding how to evaluate and analyze these functions involves considering each piece separately and ensuring the function's overall behavior is consistent with the conditions of limits and continuity. In this problem, the function f(x) is defined differently over three intervals, requiring careful examination of each segment.

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Textbook Question

Use the graph of f in the figure to do the following. <IMAGE>

a. Find the values of x in (-2,2) at which f is not continuous.

Textbook Question

Continuity of a piecewise function Let g(x) = <matrix 2x1> For what values of a is g continuous?

Textbook Question

Continuous Extension

Explain why the function ƒ(𝓍) = sin(1/𝓍) has no continuous extension to 𝓍 = 0.

Textbook Question

[Technology Exercise] In Exercises 33–36, graph the function to see whether it appears to have a continuous extension to the given point a. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at a. If the function does not appear to have a continuous extension, can it be extended to be continuous from the right or left? If so, what do you think the extended function’s value should be?

g(θ) = 5 cos θ / (4θ ― 2π) , a = π/2

Textbook Question

Limits and Continuity

On what intervals are the following functions continuous?

a. ƒ(x) = x¹/³

Textbook Question

Limits and Continuity

On what intervals are the following functions continuous?

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Textbook Question

Limits and Continuity

On what intervals are the following functions continuous?