Skip to main content
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 2.4.49
Chapter 2, Problem 2.4.49

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
1
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

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.
Recommended video:
06:21
Properties of Functions

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.
Recommended video:
05:50
One-Sided Limits

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).
Recommended video:
05:50
One-Sided Limits
Related Practice
Textbook Question

At what points are the functions in Exercises 13–30 continuous?

y = (2x – 1)¹/³

212
views
Textbook Question

At what points are the functions in Exercises 13–30 continuous?

y = √(x⁴ +1)/(1 + sin² x)

223
views
Textbook Question

Using the Formal Definition


Prove the limit statements in Exercises 37–50.


limx→−3 (x² − 9) / (x + 3) = −6

250
views
Textbook Question

Infinite Limits


Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.


lim x→−5⁻ (3x) / (2x + 10)

335
views
Textbook Question

In Exercises 77–80, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.)


lim x → ±∞ k(x) = 1, lim x → 1⁻ k(x) = ∞, and lim x → 1⁺ k(x) = −∞

289
views
Textbook Question

Suppose that a function f(x) is defined for all x in [-1,1]. Can anything be said about the existence of limx→0 f(x)? Give reasons for your answer.

376
views