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

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

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1:52 minutes

Problem 2.4.49

Textbook Question

Theory and Examples

Suppose that f is an odd function of x. Does knowing that limx→0+ f(x) = 3 tell you anything about limx→0− f(x)? Give reasons for your answer.

Verified step by step guidance

Recall the definition of an odd function: A function f(x) is odd if for all x in the domain of f, f(-x) = -f(x).

Consider the given limit: lim as x approaches 0 from the positive side (x → 0+) of f(x) is 3. This means as x gets closer to 0 from the right, f(x) approaches 3.

Use the property of odd functions: Since f is odd, f(-x) = -f(x). Therefore, if x approaches 0 from the positive side, -x approaches 0 from the negative side.

Apply the odd function property to the limit: lim as x approaches 0 from the negative side (x → 0−) of f(x) is equal to lim as x approaches 0 from the positive side of -f(x).

Conclude the relationship: Since lim as x → 0+ f(x) = 3, it follows that lim as x → 0− f(x) = -3, due to the odd function property.

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Key Concepts

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

Odd Function

An odd function is a function f(x) that satisfies the condition f(-x) = -f(x) for all x in its domain. This symmetry about the origin implies that the graph of the function is symmetric with respect to the origin. Understanding this property is crucial for analyzing the behavior of the function as x approaches zero from both the positive and negative directions.

Limit from the Right

The limit of a function as x approaches a value from the right, denoted as limx→a+ f(x), describes the behavior of the function as x gets arbitrarily close to a from values greater than a. In this context, knowing limx→0+ f(x) = 3 indicates that as x approaches zero from the positive side, the function approaches the value 3.

Limit from the Left

The limit of a function as x approaches a value from the left, denoted as limx→a− f(x), describes the behavior of the function as x gets arbitrarily close to a from values less than a. For an odd function, the limit from the left can be deduced from the limit from the right due to the symmetry property, specifically limx→0− f(x) = -limx→0+ f(x).

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Related Practice

Textbook Question

Limits as x → ∞ or x → −∞

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.

lim x→⁻∞ (³√x − ⁵√x) / (³√x + ⁵√x)

Textbook Question

Limits as x → ∞ or x → −∞

lim x→∞ √(x² + 1) / (x + 1)

Textbook Question

Limits as x → ∞ or x → −∞

lim x→∞ (x − 3) / √(4x² + 25)

Textbook Question

Theory and Examples

Once you know limx→a+ f(x) and limx→a− f(x) at an interior point of the domain of f, do you then know limx→a f(x)? Give reasons for your answer.

Textbook Question

Formal Definitions of One-Sided Limits

Greatest integer function Find (a) limx→400+ ⌊x⌋ and (b) limx→400− ⌊x⌋; then use limit definitions to verify your findings. (c) Based on your conclusions in parts (a) and (b), can you say anything about limx→400 ⌊x⌋? Give reasons for your answer.

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)

Textbook Question

[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.

a. How does the graph behave as x → 0⁺?

Give reasons for your answers.

y = (3/2)(x − (1 / x))²/³

Textbook Question

[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.

c. How does the graph behave near x = 1 and x = −1?

Give reasons for your answers.

y = (3/2)(x − (1 / x))²/³

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Theory and ExamplesSuppose that f is an odd function of x. Does knowing that limx→0+ f(x) = 3 tell you anything about limx→0− f(x)? Give reasons for your answer.

Key Concepts

Odd Function

Limit from the Right

Limit from the Left

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Theory and Examples

Suppose that f is an odd function of x. Does knowing that limx→0+ f(x) = 3 tell you anything about limx→0− f(x)? Give reasons for your answer.