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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 2.4.47
Chapter 2, Problem 2.4.47

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.

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To determine \( \lim_{x \to a} f(x) \), we need to consider both the right-hand limit \( \lim_{x \to a^+} f(x) \) and the left-hand limit \( \lim_{x \to a^-} f(x) \).
The limit \( \lim_{x \to a} f(x) \) exists if and only if both the right-hand limit and the left-hand limit exist and are equal.
If \( \lim_{x \to a^+} f(x) = L \) and \( \lim_{x \to a^-} f(x) = L \), then \( \lim_{x \to a} f(x) = L \).
If \( \lim_{x \to a^+} f(x) \neq \lim_{x \to a^-} f(x) \), then \( \lim_{x \to a} f(x) \) does not exist.
Therefore, knowing both \( \lim_{x \to a^+} f(x) \) and \( \lim_{x \to a^-} f(x) \) allows us to determine \( \lim_{x \to a} f(x) \) only if they are equal.

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Key Concepts

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

Limits from the Right and Left

In calculus, the limit of a function as it approaches a point from the right (denoted as limx→a+ f(x)) and from the left (denoted as limx→a− f(x)) are crucial for understanding the behavior of the function near that point. If both one-sided limits exist and are equal, it suggests that the overall limit at that point may also exist.
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Existence of a Limit

A limit exists at a point if the values of the function approach a specific number as the input approaches that point from both sides. For limx→a f(x) to exist, it is necessary that both limx→a+ f(x) and limx→a− f(x) not only exist but also be equal. If they differ, the overall limit does not exist.
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Continuity and Discontinuity

A function is continuous at a point if the limit as it approaches that point equals the function's value at that point. Discontinuities can arise when the one-sided limits are not equal or when either limit does not exist. Understanding these concepts helps in determining the nature of the function at the point in question.
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Related Practice
Textbook Question

Limits of Average Rates of Change


Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.


f(x) = 3x - 4, x = 2

Textbook Question

Average Rates of Change


In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.


P(θ)=θ³ − 4θ² + 5θ; [1,2]

Textbook Question

Calculating Limits


Find the limits in Exercises 11–22.


limt→6 8(t−5)(t−7)

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Textbook Question

A function value Show that the function F(x) = ( x − a)²(x − b)² + x takes on the value (a + b)² for some value of x.

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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

Infinite Limits


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


lim x→−5⁻ (3x) / (2x + 10)