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Problem 44 Medium Difficulty

For $ f(x) = \frac{2}{x} - \frac{1}{\ln x} $ find each of the following limits.

(a) $ \displaystyle \lim_{x \to \infty} f(x) $
(b) $ \displaystyle \lim_{x \to 0^+} f(x) $
(c) $ \displaystyle \lim_{x \to 1^-} f(x) $
(d) $ \displaystyle \lim_{x \to 1^+} f(x) $
(e) Use the information from parts (a) - (d) to make a rough sketch of the graph of $ f $.

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Video Transcript

This is problem number forty four of the Stuart Calculus. Eighth addition. Section two point six for F of X is equal to to Rex minus one over L. A. Next find each of the following limits part a limit is X approaches. Infinity of the function affects well as experts is infinity two over X approaches to our infinity A very large number that nominator means at this term goes to zero. Um, we have one over Alan approaching infinity. Also another example of one over a very large number, which also goes to zero. Therefore, this, um, it goes to zero. For part. Being limit is experts zero from the right suspects as experts is here from the right, we also we have a very small number here in the denominator to do at a very, very small positive number. It's very large, infinite number, Linus, as we approach zero from the right for Ellen Ah, we're approaching a very negative large negative number. Negative infinity. Which means if this term banishes zero totoro dysfunction, purchase positive infinity. As you put Sarah from the right, the limit is expert is one from the left for a fix. Well, as we approach one from the left to over one. From left, there's just two. So this first term purchase too, Linus Olympics as you approach Olympics Ah, easy approach one on this graph from the left, you're approaching zero from the left, approaching a small made of number. And as we approach is well, negative number of this ah number of beverages this affection one over small things. Remember, it is a, uh, large negative number infinity. And if we calculate this to minus, think very large negative number very large, positive number for a party. It's a limited approaches. Ah X approaches one from the right for the function act This first terms, it's just too still minus and one over. As you put one from the right for the function of Ellen, your machine zero from the rape. A very small positive number in the denominator means at this number of parties, Very large, positive, infinite, really large number. And if we take the difference between to relatively small number minus so very large number Ah, this function ever just two negative in committee. So with all this said and all this calculated can I try for a sketch of this function of where At one. It has a request in tow, since as an indefinite this continuity. This is one. And as we can see function purchase positive infinity as he approaches over on the right positive infinity as they put a warning from the left. Negative infinity is your personal infirmary and zero as we approach infinity. So this function here is a rough sketch of the function to over explain this one over Alan X and is consistent with all the limits that we calculated.

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