Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problem 2.2.8

Chapter 2, Problem 2.2.8

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.

Verified step by step guidance

Consider the definition of a limit: lim(x→c) f(x) exists if f(x) approaches a specific value L as x approaches c from both sides.

Since f(x) is defined for all x in the interval [-1, 1], it means f(x) is defined at x = 0 and in the neighborhood around x = 0.

For the limit lim(x→0) f(x) to exist, the left-hand limit (as x approaches 0 from the left) and the right-hand limit (as x approaches 0 from the right) must both exist and be equal.

However, the problem does not provide any information about the behavior of f(x) near x = 0, such as continuity or specific values, so we cannot definitively conclude whether the limit exists.

Without additional information about the behavior of f(x) near x = 0, such as continuity or specific functional values, we cannot determine the existence of lim(x→0) f(x).

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Key Concepts

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

Limit of a Function

The limit of a function describes the behavior of the function as the input approaches a particular value. In this case, we are interested in the limit as x approaches 0. A limit exists if the function approaches a specific value from both the left and right sides of that point.

Continuity

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. If f(x) is continuous at x = 0, then limx→0 f(x) = f(0). Understanding continuity helps determine if the limit exists based on the function's behavior around that point.

Existence of Limits

The existence of a limit is not guaranteed just because a function is defined in an interval. For limx→0 f(x) to exist, the left-hand limit and right-hand limit as x approaches 0 must be equal. If the function has any discontinuities or undefined behavior at or around x = 0, it may affect the existence of the limit.

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Related Practice

Textbook Question

Limits of quotients

Find the limits in Exercises 23–42.

limu→1 (u⁴ − 1)/(u³ − 1)

Textbook Question

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

y = (2x – 1)¹/³

Textbook Question

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

y = √(x⁴ +1)/(1 + sin² x)

Textbook Question

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→−3 (x² − 9) / (x + 3) = −6

Textbook Question

Use the Intermediate Value Theorem in Exercises 69–74 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.

x³ − 3x − 1 = 0

Textbook Question

In Exercises 1–4, say whether the function graphed is continuous on [−1, 3]. If not, where does it fail to be continuous and why?

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