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