Problem 75 Hard Difficulty

Discuss the asymptotic behavior of $ f(x) = (x^4 + 1)/x $ in the same manner as in Exercise 74. Then use your results to help sketch the graph of $ f $.

Answer

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Video Transcript

we want to discuss the SM topic. Behavior of F F X is equal to X to the fourth plus one over X, the same manner that we did in exercise 74. And then we want to use this to help us sketch the breath. So in 74 they give us what they believe the psychotic behavior should be. Ah, and this one we need to figure out what we think it might be. So if we were to first divide X into exile before and one, we would end up with X cubed plus one over X. Now, if you look at this well, we know that X cubed is going to tend to plus or minus infinity, and we know that one over X is going to tend to zero as X goes to plus or minus infinity. So as X goes to plus or minus infinity so essentially, for very large values, this one over X doesn't really matter. And this X cubed is a thing that truly matters. So what we want to show is that the limit as X approaches, plus or minus infinity of X cubed are not execute but X to the fourth plus one over X minus X cubed is equal to zero. So we want to show this because if this is the case, then we know that the end behavior for our function f of X really is exe cute. So let's go ahead and start with that. So let's just simplify the inside first. So I'll go ahead and do that up here. So we have X to the fourth plus one over X minus x cute. Now we need to get a common denominator. So let's move by Execute top and bottom by X simplifying this down. This is going to give us X to the fourth plus one minus X to the fourth all over X and well, now those extra force cancel out and we just be left with one over X. So all we need to do is show that now the limit as X approaches infinity of one of her ex Is he good zero? Well, that's the century will be said here at the very start. So that zero and then likewise as X approaches negative infinity of one of Rex. Well, that would also be zero. So we show that this year simplifies down too. One of these cases and both of the limits end up giving us zero. So we know the end behavior of this function. Extra four plus one over X should be very close to execute. All right, now that we have that information, let's just go ahead and find our intercepts at our vertical aspirin tubes. So we I know that our ex intercepts deal with up here. So this is the ex Intercepts will be set that equal zero so he'd have X to the fourth plus one is zero. Subtract one over. We get X to the fourth is equal to negative one. Take the fourth route on each side so we get plus or minus +34 through of negative one. But this year is in imaginary number since we're taking it even root of a negative number. So that means we have no ex intercepts and then to find our horizontal Assam totes, remember or not horizontal vertical are vertical ascent oats stills with setting the denominator equal to zero. So we're going to do X is equal to zero, so that seems pretty straight or nothing to really solved for right now, let's use this information or the next part. So let's just write down right now. All we know is that we have a vertical awesome toe at X is equal to zero, and we have no ex intercepts. So let's go ahead and put down are dotted line for vertical asked him to hear. So this is at X is equal to zero. And now what we are gonna have to do is determine what side of this we need to start. So we need to look at. So we had extra the fourth plus one over X. So let's look at the limit of this as we approach zero from the right. So the limits the limit as X approaches zero from the right. So we're gonna have zero from the right to the fourth power, plus one over zero from the right. Well, if we raise anything that's slightly positive, which means coming from the right to and even power, that should still be positive. So we're going to have zero from the right, plus one over zero from the right and then adding something slightly to the right of one of zero and one well that should still give us something positive and then dividing that by something positive gives us infinity. So we know that the right hand side of this will look something kind of like this. So it was going to start coming from positivity. And now this is when we use the fact that we know this has asked Antarctic behavior like X cubed, so is going to have to rebound up and then start to fall this green line up to positive infinity. And then we could do the same thing to find the limit as exit purchaser from left. Or we can use the fact that this is like one over X and we know that one of Rex has opposite in behavior. So that means we're gonna have to start from negative infinity over here, and then we're gonna just keep on going up. There's no X intercepts or anything. So then it's just going to turn around and then start following this green line like that. So this is a nice way for us to go about getting a sketch of the graph without doing everything that we've done in this chapter so far,