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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} $

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

Missouri State University

University of Michigan - Ann Arbor

University of Nottingham

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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Find the horizontal and ve…

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$41-46$ Find the horizonta…

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

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