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In Exercises 89-94, determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If $\lim f(x)$ exists, then $f$ is defined at $x=a$.

Name: Problem 89, Chapter 9: Applied Mathematics: For the Managerial, Life, and Social Sciences
Uploaded: 2020-11-20T16:39:18-08:00
Duration: 1 min 45 s
Channel: Stephen Hobbs
Description: In Exercises 89-94, determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If limf(x) exists, then f is defined at x=a.

   In Exercises 89-94, determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false.
If $\lim f(x)$ exists, then $f$ is defined at $x=a$.

Applied Mathematics: For the Managerial, Life, and Social Sciences

Soo T. Tan 5th Edition

Chapter 9, Problem 89

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Solved by Expert Stephen Hobbs

11/20/2020

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If $\lim f(x)$ exists, it means that the limit of $f(x)$ as $x$ approaches some value exists. This means that as $x$ gets arbitrarily close to that value, $f(x)$ also gets arbitrarily close to a certain value. Show more…

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In Exercises 89-94, determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If $\lim f(x)$ exists, then $f$ is defined at $x=a$.

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Key Concepts

Limit of a Function

The limit of a function as x approaches a certain value describes the behavior of the function in the vicinity of that point. Importantly, it does not depend on the function's value exactly at that point, but rather on the values arbitrarily close to it.

Definition of a Function at a Point

A function being defined at a point means that the function has an explicit assigned value at that specific point. The existence of a limit as x approaches that point does not require that the function is defined there, which can lead to situations where a function has a limit at a point of discontinuity.

Removable Discontinuity

A removable discontinuity occurs when a function is not defined at a point or is defined differently from the limit value, even though the limit of the function exists as x approaches that point. This concept illustrates that the existence of a limit does not necessarily imply that the function is defined at the limit point.

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