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University of North Texas

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Problem 71 Hard Difficulty

Show that the curve $ y = x - \tan^{-1} x $ has two slant asymptotes: $ y = x + \pi /2 $ and $ y = x - \pi /2 $. Use this fact to help sketch the curve.

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Video Transcript

we want to show that the girl winds into X minus tangent. Inverse of X has to sign some tips being it. Why is it two X plus or minus pi half And then we want to use the's s and hopes to help us sketch the curve. Now, remember to show something has come to what we're going to want to show. Is that the limit? As X approaches plus or minus infinity of F of X minus G of X is equal to zero where this function G of X would end up being our slapped right. So let's go ahead and first get some inspiration as to which of these we should use for plus and minus infinity. So we know the limit as X approaches Infinity of XO the limit as experts, if any of ex Well, this goes to infinity, and likewise or negative. This goes to negative infinity. So that doesn't really give us anything but tangent. Inverse. So the limit as X approaches tangent inverse of x. Well, we know that this year approaches pi half and we know the limit as X approaches Negative. So negative infinity of inverse tangent would be negative pie, huh? So given this information, we might suspect that as why tends towards infinity. So our original function here it would go to X minus pi. How and then as it tends to negative infinity, it would go to X by nous negative pie half or x plus pie half. So let's go ahead and show that those two sly ass symptoms are actually the case. All right, so we want to show first the limit as ex purchase infinity of so X minus tangent and verse X. And then we're going to do minus X minus pi, huh? So here the exes will cancel out and we'll be left with the limit as X approaches Infinity of negative tangent inverse x plus pie hat because this negative here gets distributed to that. And then we already said that the limit has experts Divinity of tension Unversity pie, huh? So we get negative pie half plus high half, which is zero so that one checks out for being our slight. Ask himto as this goats towards infinity Now to show the other one works. So maybe I'll write this over here. So x minus tangent inverse of X will that tends towards X minus pi half. I know we could do the same thing for the other one. So the limit as X approaches Negative. Infinity of X minus tension in verse. Specs minus X plus. Hi. How well again. The exes here will counsel out, and then we can go ahead and let's just apply the limit right away. So the point X that we have left is right here, so that would become negative. Negative pie half. So we have negative negative pie half, and then distributing this negative to that would give us minus pi half. So these two negatives council Well, that we have pie half minus by half, which gives us zero. So that is also a slack past two. So actually I should put This is for infinity right there and then our other one. So x minus. Tanja inverse of X this year will tend towards X plus pie half when x is going too negative. Infinity. Now let's go ahead. And one other thing we might want to do if we're gonna graft this is plot our intercept. Let's do that over here in this bottom left corner. So we want to let X equal to zero. So we're gonna have why is equal to zero minus tangent. Inverse of zero. Well, Tanja inverse of zero is gonna be zeros. We get zero minus zero or just zero, so we know that the origin will be one of our points on our graph. Now, I went ahead and plotted the to slap ass in tips here, and so we know that 00 is a point. So let's go ahead and graft that really quickly. So that's going to be right here. And now we know as Ex tends towards infinity, we should tend towards X minus pi half. So the graphs should look something kind of like this getting really close to the red line. And then we also know that as extends towards negative infinity dysfunction should get really close to X plus pie half or this blue line. So this is how we could go ahead and just get a sketch of this graph without doing too much work using our to slap pass in twos