Problem 72 Hard Difficulty

For the limit $$ \lim_{x \to \infty} \frac{1 - 3x}{\sqrt{x^2 + 1}} = -3 $$ illustrate Definition 7 by finding values of $ N $ that correspond to $ \varepsilon = 0.1 $ and $ \varepsilon = 0.05 $.

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Video Transcript

This is problem number seventy two of the sewer calculus eighth edition, section two point six for the limit. The limit is ex purchase infinity one minus three X divided by square root of the quantity X squared plus one equals three. Kill a street definition seven By finding values bin they correspond to absolutely cools to Sierra point one and absolutely equals to zero point zero fighters. And our definition, it has to do with defining there that the difference between the function and its limit the distance between the two or the absolute value be less than the value Absalon provided that you have an ex and greater than a certain value and right to the school together. And in our case, this is our function If and this is our limit own All right. So we need to find about you end that essentially guarantees of the function is less than Absalon away from the limit. So take a look at this function and we plant this function here one minus three x divided by the square root of the quantity X squared plus one. Ah, and we see that dysfunction approaches approaches Negative three. But we know that as the limit is, X approaches. Infinity Little, many calls native. Three. That means it'LL person ing into three but never be equal to negative three. So, really, when we're solving this problem here that we want the function to be within point one of its limit. Ah, that's exactly what determines our in. So within point one of negative three Centenary goes beyond negative three or equals negative three within point one means negative two point nine of negative three Thank you two point. And is Sarah quaint one away from negative three. So you need a next value at least equal to eleven point three to be able to achieve a value on the function within a zero point one of the limit, which is equal to make it three. So for this first step, Swan Lun is eleven point three Now. As we hit more precise with our Absalon point o five, we expect the larger Valley event how much more strict, stricter value. And that's exactly what we've seen. The function. It's causing negative three as we continued to the extraction and we need an X X value at least equal to twenty one point four In order to be within a point, there are five over the limit. So for this Absalon, our numbers approximately twenty one point four.