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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1:21 minutes

Problem 2.5.5b

Textbook Question

Exercises 5–10 refer to the function
f(x) = { x² − 1, −1 ≤ x < 0
2x, 0 < x < 1
1, x = 1
−2x + 4, 1 < x < 2
0, 2 < x < 3
graphed in the accompanying figure.
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b. Does lim x → −1⁺ f (x) exist?

Verified step by step guidance

Identify the piece of the piecewise function that is relevant for x approaching -1 from the right. Since we are interested in the limit as x approaches -1 from the right (x → -1⁺), we need to consider the interval that includes values just greater than -1.

Examine the piece of the function that applies to the interval -1 ≤ x < 0. In this interval, the function is defined as f(x) = x² - 1.

To find the limit as x approaches -1 from the right, evaluate the expression x² - 1 as x gets closer to -1 from values greater than -1.

Substitute values that are slightly greater than -1 into the expression x² - 1 to observe the behavior of the function. For example, consider values like -0.9, -0.99, etc., and calculate the corresponding f(x) values.

Conclude whether the limit exists by determining if the values of f(x) approach a specific number as x approaches -1 from the right. If they do, the limit exists and is equal to that number.

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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 as the input gets arbitrarily close to a specific point, which is crucial for understanding continuity and differentiability.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. Understanding how to evaluate piecewise functions is essential for analyzing their behavior at specific points, especially at boundaries where the definition of the function changes.

Right-Hand Limit

The right-hand limit of a function at a point is the value that the function approaches as the input approaches that point from the right. This concept is particularly important when evaluating limits at points where the function's definition changes, as it helps determine if the limit exists and what value it approaches.