Is f(x)={ (sin x)/(|x|), x ≠ 0 1, x=0 . continuous at x=0 ? Explain.

Name: Problem 52, Chapter 1: Calculus Early Transcendentals
Uploaded: 2020-08-18T04:46:13-08:00
Duration: 1 min 17 s
Channel: R M
Description: Is f(x)={ (sin x)/(|x|), x ≠ 0 1, x=0 . continuous at x=0 ? Explain.

step 1 problem number 52 so f of x we can define f of x as follow sign x over x when x is bigger than 0

Is f(x)={ (sin x)/(|x|), x ≠ 0 1, x=0 ....

Key Concepts

Piecewise-Defined Functions

Piecewise-defined functions use different expressions to define the function over different parts of the domain. Analyzing the continuity of such functions involves checking the consistency of the function's behavior at the points where the definition changes.

One-Sided Limits

One-sided limits involve the evaluation of a function as the variable approaches a point from one side—either from the left or from the right. They are essential when dealing with functions defined differently on either side of the point, as they reveal whether the overall two-sided limit exists.

Absolute Value

The absolute value function affects expressions by altering signs depending on whether the input is positive or negative. Its presence in a function can result in different behaviors on either side of a particular point, which is significant when evaluating limits and continuity.

Continuity

Continuity at a point requires that the function is defined at that point, that the limit exists as the variable approaches the point, and that the limit is equal to the function's value at that point. This concept is critical in analysis to ensure that there are no breaks or jumps in the function's graph at that specific point.

Limits

The concept of limits involves studying the behavior of a function as the input value approaches a particular point. When determining continuity, one must compute the limit from both sides of the point in question and verify that these limits coincide with the function's value at that point.

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