Problem 50 Easy Difficulty

Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.

$ y = \dfrac{1 + x^4}{x^2 - x^4} $

Answer

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Video Transcript

this problem or fifty of the Stuart Calculus eighth edition section two point six Find the horizontal and vertical hasn't OTS of each curve. If you have a graphing device, check your work paragraph in the curve and estimated master totes. The curve in this case is right equals one plus X to the fourth, divided by the quantity X squared minus X to the force. Heretical Assam dotes. We confined by setting the denominator equal to zero in this case X squared minus six of the fourth, equal to zero. We can affect her out and X squared, leaving us with one minus x squared. And we can fact throughout this difference of squares what's one's greater menace? X squared Kim affected as one plus x one minus x. Therefore, we have three X values that make this statement true. Zero negative one and positive one. The's expellees make the denominator equals zero, which means that this function is undefined at these valleys. Therefore, X equals zero X equals theta one X equals one are all vertical aspirin totes our horizontal as stoats are founded by taking the limit as X approaches infinity of the given function and or little too. To simplify this limit is to divide each term, buy eggs to the fourth. So I think it is this one of rakes in the fourth class one ratifying one over X squared minus one. And as we take the limit as express infinity, each of these terms vanish. Notice that this is the same case for ex purchase negative Infinity. He's good, very small and are a negligible compared to the other terms, such that this limited to negative one. And since we get the same answer for the limited express, infinity, as we do with experts, is negative. Infinity. There's only one way to cross him, too. Are horizontal Hasn't too equal to y equals negative. One two. We have three vertical ascent oats. Why? Why? Cause one is the only horizontal as into, And by crafting this function here, we can see the function one plus x to the fourth over, X squared minus six. Before it's part of it in red, we can see that it indeed does. After he broke a question to its At X equals negative one X equals zero. Mexico's one and one course onto Jacinto at one equals negative one