Problem

Make a conjecture about the equations of horizont…

02:00

Problem 54 Hard Difficulty

Make a conjecture about the equations of horizontal asymptotes, if any, by graphing the equation with a graphing utility; then check your answer using L'Hôpital's rule.
$y=x-\ln \left(1+2 e^{x}\right)$

Answer

$$y=-\ln (2)$$

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Calculus 1 / AB

Calculus Early Transcendentals

Chapter 3

TOPICS IN DIFFERENTIATION

Section 6

L Hopital's Rule; Indeterminate Forms

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Video Transcript

So in this problem, we're looking for the horizontal asthma totes. And when we go ahead and graph the function, you get a graph that looks kind of like this, um, me people. This that's negative. Five. This is five. Um, and it kind of looks from the picture like it's going to a number That's, um, a little less than one. So I'm going to guess that it goes to or a little bit greater than one. Um, I'm going to guess that it goes to e divided by two. That's my conjecture. Now we'll see from the work that this is actually this is actually wrong that that that is not the correct conjecture. Um And okay, so let's go ahead and do this limit as X goes to infinity of X minus natural log off one plus Teoh he x What we conduce. So is right X in a different way. We can write it as natural log of e to the X on my it's natural log one plus to eat the X. Let's go ahead and use our natural log property so we hav e to the X divided by one plus two e to the X. And now let's go ahead and use one of our, um Well, let's go ahead and use the limit Interchange property that allows us to swap the natural log with the limit. Okay. And so now, with this limit, we can go ahead and use low petals. Rule hotels rule, I guess. Let me go ahead and write that below here. So eat E to the X right by two e to the X the TX cancel and are left with natural log off. Natural log of one half. So, um, I realize I made a mistake here. I meant to say that the my conjecture that it's negative e over to left off a negative sign. Sorry about that. Anyway, let's simplify this to the correct value. So that gives us a negative natural log of to. Okay, so we know that one of the asthma totes is y equals negative natural log of two. And the question is, is there another asthma tote? Well, looking at the graph, it looks like as we approach the pot as we approach negative infinity, it doesn't take on any sort of limits. So there's only the one aspect of But let's let's be sure and consider the limits as X goes to negative infinity. Okay. Now, uhm, I'm going to skip some steps here because they're identical to the steps above. We end up with natural log of E to the X one minus to eat to the X. And now here were able to evaluate this limit because X is going to negative infinity. So we get Zeer and so So we we end up getting, um, well inside to the natural log. We end up getting something that tends to zero. So, um, with the national log of zero that's tending to negative infinity. And that's consistent with what we saw on the graph here. So there is only one horizontal asthma tote. That's that.

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