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this problem Number seventy five of this tour. Calculus
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. Eighth edition, Section two point six, part
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eight. How large do we have to take X
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so that one over X squared is less than zero
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point zero zero zero one for this part? We're
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going ahead and going to solve this outbreak. Lee
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s. So when is this true? One over
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X squared. Less than point zero zero there.
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One that's the same Assane one over X squared.
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Ah, less than ten thousand or one over ten
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thousand. Now our next step will be to take
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the reciprocal of Kotite's changing this the direction of the
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inequality. Ah, and finally we take a square
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. So in this case, we need to take
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X as large as one hundred. In order for
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us to confirm that one over X squared is less
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than a one one one ten thousand X must be
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larger than one hundred. Prepare B. Yeah,
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according to deer and five we take, our is
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equal to two. So that means the exponent on
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the X and the denominator is too. We have
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this statement the limit as ex purchase infinity of the
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function one over X squared is equal to zero.
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Prove this circle using definition seven definition seven says if
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X is greater than the value and then the distance
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or the difference between the function on the limit will
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be less than a certain no margin. A certain
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error Absalon came. So we need to be able
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to confirm this is true for any choice of Absalon
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that we can find a certain end value s so
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we can proceed and use what function we have here
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. One over x squared, the limit l which
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is zero in this case less than Absalon. We're
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trying to figure out maybe what absolute should be.
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We're going to restrict Absalon to only positive values,
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which means that can disregard the absolutely sign. Ah
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, we can take the reciprocal X squared greater than
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Absalon and ah, or ex ex Greg greater than
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one over. Absolutely. And then finally the square
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root ex greater than one over the square root of
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Absalom. At this point, this is what satisfies
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this second part. The absolute value, the quantity
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effects, F X minus l being less in a
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quantity. Absalon. We also see that this has
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to be true for X is greater than a certain
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number. And so here we conclude that if we
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choose and to be at least one over the square
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root of Absalon Ah, then that confirms the statement
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. So for any Absalon, we can choose ups
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on to be one and choose Epsilon to be point
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one point zero zero one every single time. Well
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, we will be able to choose an appropriate and
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value such that this statement is true and according to
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definition seven, because that is true. This limit
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is in fact, true that the limited his expressions
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affinity of one over X squared is equal to zero
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.