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By comparing Definitions 2, 3, and 4, prove Theorem 2.3.1.

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 4

The Precise Definition of a Limit

Limits

Derivatives

Missouri State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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.

03:54

By comparing Definitions $…

01:05

Prove statements 3 and 4 o…

01:48

This is problem number forty of the Stuart Calculus eighth edition, Section two point for By comparing definitions, two, three and four prove the serum to point three point one. They're, um, two point three point one state step the limit from for exporting a from the left of F, if that is equal to a limit. And Oh, and they've filament as experts is a from the right of that same function, it also equal at all. Then the limit from the left equals elimination. Right? And this the entire limited true. And we're going to show that using these three definitions here definitions two, three, and four from this two point four section. So we're gonna begin and just assumed on a definition three initiative definition for our true so poor definition. Three. We're assuming that the Limited Express today from the left is are equal to a felon for these conditions, and we're going to just write this condition here. Um, that eh? Minus delta and worry McCall. This Delta one is less than X. Just a name. Um, since the limit is true, then this is also true. Came Well, this is from definition three. If we also state session definition poor is true and given round. We get something similar to this history from four. We can't thes this condition. And for a cz less than X is list than a plus. Another Toto he's talked to, uh then, well, you get the same absolute value of the difference between the function and the limit. So we're going to work with these two inequalities. Um, and we have to deal with this potentially different Delta. One daughter too, are nowhere in the state of you. Talkto is equal to the minimum because that's always important, making sure we have the smallest, the minimum either Delta one our doctor to whichever one is smaller. And if we do that, then and we can essentially ah, name a new range one that comes from the first inequality a minus. Just tell, too, since we already chose the minimum. This is definitely less than X, which is something less than a and we continue from here since the's should combine hes deaf definitely listen x, which is definitely less than earn it. This is one side at the other side would be that is less an ex which is less than a with TelDta again, we replaced it with the minimum delta. Combining these two gives us a minus Delta. That's the next listen a plus Delta. And now we have our range That is consistent with this statement. Yeah, Either these statements were the same. We can rearrange this subject a from each term and rewrite this ands. The absolute value of the difference between next name is less than the value delta, which we already confirmed. It's the minimum of the two original taught isn't Dimensions two hundred. And like we said, this no still come terms at the different scene. The function, Um since those air to those both lead to the fact that this definition is true for definition, too. Thus, this is proven here. The limited this purchase a completely other after a little. And this is exactly the serum to point three point one. That states depth limit from the left equals left from the right then the entire limit as exposes equal to that same limit, Ellen. And thus we have proven theorem two point three point one

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