Show that |(2+x)^2-4| ≤5|x| if -1<x<1 . Use this to show that limx →0(2+x)^2=4

Name: Problem 36, Chapter 1: Calculus, Early Transcendentals
Uploaded: 2021-12-13T05:34:55-08:00
Duration: 5 min 5 s
Channel: Harshita Goel
Description: Show that |(2+x)^2-4| ≤5|x| if -1<x<1 . Use this to show that limx →0(2+x)^2=4

step 1 In this problem of limit and continuity we have to show that modulars of 2 plus x square this is

Show that |(2+x)^2-4| ≤5|x| if -1<x<1 . Use this to show...

Key Concepts

Algebraic Manipulation and Factorization

In this problem, algebraic manipulation, particularly factorization, is used to rewrite expressions in a more manageable form. By expanding or factoring expressions, one can simplify differences such as (2+x)² - 4, making it easier to analyze their behavior. This is a fundamental technique in mathematics that helps in isolating variables and studying limits, inequalities, and other properties of functions.

Absolute Value Inequalities

Absolute value inequalities are crucial for bounding expressions and understanding how one expression behaves relative to another. In the context of the problem, establishing an inequality like |(2+x)² - 4| ? 5|x| helps provide a clear measure of how close the function value is to the limiting value as x approaches zero. This concept is vital in analysis for proving convergence and continuity by controlling error margins.

Limit of a Function and Epsilon-Delta Definition

The limit of a function, especially when defined via the epsilon-delta approach, is a cornerstone of mathematical analysis. It formalizes the intuitive notion of a function's behavior as its input approaches a particular point. In this problem, after establishing the inequality, one shows that as x tends to zero, the function (2+x)² tends to 4, which is a specific instance of using the epsilon-delta definition to prove limits, thereby confirming the continuity of the function at that point.

Transcript

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