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University of North Texas

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Problem 76 Hard Difficulty

Use the asymptotic behavior of $ f(x) = \sin x + e^{-x} $ to sketch its graph without going through the curve-sketching procedure of this section.

Answer

see graph

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Video Transcript

We want to use the same topic behavior of sign of X plus e to the negative X to go ahead and help us sketch this graph without going through the whole process that we've done destructor so far. So to get some inspiration as to what the s and tonic behavior of this might be, let's go ahead and take the limit of each of these functions individually so we can get an idea of where each of them would go. So let's do the limit as X approaches Infinity of sign of X, Will we no sign of ex continuously all sleaze between one and negative one, so it never gets closer to anything. So we know that this limit is not going to exist and we can say the same thing or the negative infinity is well so for plus or minus infinity, Sign of X doesn't do anything. Now we know the limit as X approaches infinity of e to the negative X Well, this here would go to zero because remember e to the negative X This is really one overeat X. So we have something going to infinity and dividing that into one or constant, so that should go to Syria. And now let's check for the other side. So the limit as X approaches negative infinity of e to the negative X Well, this here would be the same as the limit as X approaches infinity of e to the X and we'll eat of X goes to positive infinity. So if we were to maybe make a guess so the limit as X approaches infinity of f of x, we might make the guests that this goes to sign of ex Since we found that e to the negative X goes to zero in this case and the limit as X approaches negative infinity of f of X. We would suspect that would go to positive infinity since sign of exit Just Balan between negative one and one and e to the negative x goes to infinity. Well, we can at least show this one has that slack ass in tow by doing what we've done in some of the previous problems in this chapter. And remember to do that what we want to show. We want to show that the limit as X approaches infinity of f of X minus sign of X. We want to show that this is equal to zero. So if this is the case, then we know the behavior as it goes to infinity will be sign of X. So let's go ahead and plug everything in. So have the limit as X approaches infinity. Oh, well, F of X was a sign of X must heed to the negative X minus side of X, the sign of X's Cancel out with each other. Then we have the limit as X approaches infinity of e to the negative x and again that zero So that one checks out. So we know as it goes to infinity, it will be a sign of X. So now let's go ahead and use this information to help us craft. And one other thing that we should probably do before. Let's find f zero just so we have an idea of where we're starting, so I'm gonna have f zero is equal to We'll sign of zero, which is zero and then e to the negative zero would just be each zero or what? So f of zero is one all right, so we can go ahead in plot. That there. And now we know to the left of this this is just supposed to go off to positive infinity and to the right dysfunction should get really close to war. I is able to sign of X, which I have graft here and read already. So it probably comes up like this goes up a little bit and then it gets really close to this red line and just kind of wiggles are rounded in some sense like that. And this is how we could go about getting at least a sketch of the graph without going into too much Ditto.