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(a) Find the vertical and horizontal asymptotes.

(b) Find the intervals of increase or decrease.

(c) Find the local maximum and minimum values.

(d) Find the intervals of concavity and the inflection points.

(e) Use the information from parts $ (a) - (d) $ to sketch the graph of $ f $.

$ f(x) = \frac{e^x}{1 - e^x} $

a) Vertical Asymptote $x=0 ;$ Horizontal Asymptotes $y=-1,0$

b) Increasing: $(-\infty, 0)(0, \infty)$

c) Use critical points from part b

d) No point of inflection

e) SEE GRAPH

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Missouri State University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

Okay, So you're being asked to find the very corners on asking Tom affect, so Ah, your coin two first, try to find that there is a bit of class and so so critical acid. It occurs when the denominator is equal to deal. So one minute you too, dear. Because you're also added together. Actually get a boner when you put it back and you plug in zero, or you could just Okay, so there's two ways to do it. You can take the natural laws on both sides and then desperately I want this eliminates toe e and national level one zero. So this will be zero is equal to X. So we have a vertical. Yeah, and X equals zero. Now to find the horizontal ass until we take the limit. As X goes to infinity s o, we first find it an infinity. So that would be it. The X over one, minus eighty react you, Khun, do direct substitution. I'm not your ex amputation. So, um, we're gonna pull out a fact. Evita Deac, toe limit of X goes to infinity. You pull out ttx that this will be one and then you do it director here again. So this will be one Overeating, Jack minus one. And then you cancel your Zodiac. And then if you take the Limited's exclusive infinity, this will become Ah, So this will be one over and then one over infinity and zero zero minus ones. This will be negative form. Now, let's take the limit as X goes to negative in Trinity because we'LL be eating the egg over one minute into the uh, for this one, you have to apply. Ah, let's see. You can apply a sort of same principle, but you can think of the graft to remember that the graph of Italy it looks like this, so it it rises. Looks something like this. So it goes off to infinity. But as it goes to negative, infinity approaches zero. So whenever you see limited expression and infinity of ttx, you could assume that easy except zero. So this Khun b AA zero over one minus zero and this will equal zero. So this was really important to have an idea of the shape of some of the important functions the shape of the graph, and it really helps. You were finding that limit when it comes to things like this to make it more intuit intuitive. So we have two horizontal asking me have won at wife with negative point, you know, and waggles Knight of one. No, the fine where the function of increasing the solution flight thie first river after So we find the first derivative You popped the question rule and you get to the back over one minded. You too, Jack squared and way said, this equals zero Andi supposed to find the critical numbers of this function. But since into the extra nautical zero, we have to look for places where this function is undefined. So when the bottom decoders also one minus into the act squared equals here. So you get out of the school by twenty per sciences have become one of us because it acts of desecration, X equals zero and then you do a sign tried evaluation. So add zero. The sign of prime the worth less and zero. It is actually a positive, and if it's great into, it was positive as well. So this function is always increasing so increasing from negative infinity to infinity. No local men or max or no local men or Mac Ah, To find where the function and find the functions Con cavity would take the thing and do every day. This turned out to be e to the X Class e Teo over one minus into the cube. So the secret hero, the talk cannot equals, you know, it is impossible. So we look at where the denominator Because you're also one minutes, Q Yeah. The Cuba taking cube roots. So the warning calls each X again and now we actually calls, you know. So again, we evaluate around zero We were assigned charge. Sorry, that is not I'm straight. Ah, Teo. So keep looking. Value less angio into F double crime. Okay, you get positive numbers and you put greater than doing it. Negative number. So it's Kong gave up and down and we have inflection point at X equals you. So now we have enough information to draw the graph of our function. So we have the function first of all, as it goes to negative infinity approaches zero. So it will come from the road. No. And then it is Khan cave up, So it's going to increase and then I'll go off to infinity because X equals, you know, the vertical Assam toe. And then he said, it is conclave down, but it is also increasing. So the only way to be conquered down an increasing It has to look like this. And we have a horizontal ass in tow at Michael. Negative one. And that makes sense. So it would come up like this. And that is the graph. What I think.