### Using the $\varepsilon$ -\delta Definition of Lim…

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Problem 46

Using the $\varepsilon$ -\delta Definition of Limit In Exercises $45-56$ , find the limit $L$ . Then use the $\varepsilon-\delta$ definition to prove that the limit is $L$ .
$$\lim _{x \rightarrow-2}(4 x+5)$$

$$\quad \delta=\frac{\epsilon}{4}$$

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## Video Transcript

Yeah, In this problem, we're asked to find the limit and sex approaches Negative two of four X. You think that Absalon Delta definition of a limit, which we see graphically to the left? We can deduce that the women out is equal to make it 30. Now we want to write a proof using he Absalon Delta definition of limit as well. In other words, we want to show that for every Absalon greater than zero, there exists a delta greater than zero such that ab flex, which is for experts five minus negative. Three absolute value is less than epsilon. Whenever zero is less than the absolute value of X minus negative too, is less than doubt him or when we plug in of the vax and simplify both of these such that the absolute value of four experts eight is less than epsilon. Whenever zero is less than the absolute value of X, plus two is Liston. So we wantto find the appropriate that you freed up him. So first off, let's note that Yep. So value of four x plus eight deceitful two, four times X plus two in absolute you. So we can kind of the factory before there, which then we can rewrite as thie ops volume four times the absolute value of X plus two. Buy properties of absolute values with any kind of, um, take the absolute value of the product, and that will be equal to quite a bit of advice. Um, so we know that Forrest positive, so we can actually write the absolute value for as simply four times the value of X plus two. Now, note that we have a relationship between this term here. You kind of were in it this way and this bound on delta. So if we let Absalon or sorry, if we let Delta equal to Absalon, divided by this number here for then we have the absolute value for X Plus eight just last. Then Absalon and we are done.