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Problem 16 Easy Difficulty

Prove the statement using the $ \varepsilon $, $ \delta $ definition of a limit and illustrate with a diagram like Figure 9.

$ \displaystyle \lim_{x \to 4}(2x - 5) = 3 $


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Video Transcript

This is problem number sixteen of the Stuart Calculus eighth edition Section two point four. Prove the statement using the absolute Delta definition of a limit and illustrate with the diagram. Like figure nine, the limit is ex a purchase for of the quantity to X minus five. Has he called the three and our Absalon Delta definition of a limit says that this limit exists and that this limit is equal to three for every as long as there's a every Absalon greater than two. That there is a delta greater than zero such that if the difference between excellent four lesson Delta, that's the value of that difference. Ah, then the absolutely of the difference between the function and three in this case, the limit is less than epsilon. So these are the conditions. Using the absolute Delta definition, we're going to use our function to X minus five and played in the street into here, so we have to expense. Five. This is the function minus the limit, which is equal to three toe minus three less than absolute two X minus five minutes. Three is the same as to explain. It's eight within the absolutely signs we're going to factor out a number two and finally divided by two on both sides. That gives us T quantity expense for the absolutely of that is less than Absalon over, too. And we recognized that this statement is the same or similar to this statement, the condition of Delta. And so we conclude the Delta, sir Ah, equals absolutely, too. That's a limiting value. But this proves that for any absolute greater than zero, there exists a delta greater than zero that satisfies both of these inequalities. And we have proven the statement based on this definition and the figure tour right I also serves to help with our Prue are proof we have plotted and blew the function to expense. Five. So it's this line here but the positive slope of two. And the diagram is just going to show that let's say that we chosing a value greater than three three plus epsilon and the right lesson three three minus epsilon at these correspond to for manis tell too, and four plus Stilton. And we see that for every Absalon that is given, there exists a delta which provides arrange that as we approach X equals for the limited purchased three and that prisoner statement

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