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

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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