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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 2.4.49
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.

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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.

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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.
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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.
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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).
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Related Practice
Textbook Question

Domains and Asymptotes


Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.


y = (√(x² + 4)) / x

Textbook Question

Infinite Limits


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


lim x→0 (−1) / (x² (x + 1))

Textbook Question

Finding Deltas Algebraically


Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.


f(x) = 1/x, L = 1/4, c = 4, ε = 0.05

Textbook Question

Slope of a Curve at a Point


In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.


y=x³−3x²+4, P(2,0)

Textbook Question

Limits of Rational Functions


In Exercises 13–22, find the limit of each rational function (a) as x → ∞ and (b) as x → −∞. Write ∞ or −∞ where appropriate.


f(x) = (2x + 3)/(5x + 7)

Textbook Question

Removable discontinuity Give an example of a function f (x) that is continuous for all values of x except x = 2, where it has a removable discontinuity. Explain how you know that f is discontinuous at x = 2, and how you know the discontinuity is removable.