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Prove the statement using the $ \varepsilon $, $ \delta $ definition of a limit.

$ \displaystyle \lim_{x \to a} x = a $

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 4

The Precise Definition of a Limit

Limits

Derivatives

Campbell University

Harvey Mudd College

University of Nottingham

Idaho State University

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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Prove the statement using …

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this problem. Number twenty three of the Stuart Calculus Ease Edition Section two point four. Prove this statement using the Epsilon Delta definition of a limit. The limit is expert is a of X is equal time. The AB signed out. The dimensional limit is summarized the right For every Absalon greater than zero there must be a delta greater than zero Such sort of the absolutely of the difference between X and aim Ms Liston daughter then the absolutely of the different between FX and own it must be lesson Absalon, if all the conditions are meant than the limit is a proven statement. So what we do is we're going to take Thesiger and Inequality and just use our function F which is in this case x minus l, which is a limit in this case. The limit is equal to a and said this less than Absalon. We also want to work with our first inequality. Experts say a being the value that this X is approaching. Should this problems, eh? His lesson, doctor, Comparing these two inequalities that we just determined we see that Delta equals Absalon is an appropriate choice. In order for these two to be consistent, right? If this first inequalities true, Delta's equal to Absalon. That makes this second equality, too. And we can choose any absolute now. And we have a Delta equal tips line. So there is a delta that agrees with any choice of song, and as such, all the conditions are meant and the statement and this limit is proof.

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