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(a) Find the $x$ -intercept(s); (b) Find the vertical asymptotes; (c) Find the horizontal asymptotes. (d) Sketch the graph.$$f(x)=x /\left(1+x^{2}\right)$$

(a) $x=0$(b) none(c) $y=0$$(d)$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 4

Limits at Infinity, Infinite Limits and Asymptotes

Derivatives

Missouri State University

Campbell University

Harvey Mudd College

Idaho State University

Lectures

04:40

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.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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(a) Find the $x$ -intercep…

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(a) Determine the $x$ -int…

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we were walking through finding the X intercept and vertical and horizontal assume totes so that we can hopefully sketch the function X divided by one plus X squared. So starting with part A here, finding this X intercept. We are familiar with the fact that we just have to set our entire function equal to zero and then solve for X. So here we'll just multiply both sides by the denominator to get rid of it. And you can see that will be left with X is equal to zero for our X intercept. So we can go ahead and just plot that right away on our craft. We're just giving ourselves basically an outline to sketch this part B. We want this vertical ascent tote and when it comes to vertical ascent oats, we know that the denominator is what we're really interested in. So we can just take one plus X squared. We're gonna set it equal to zero, solving for X. Well, uh subtract one from both sides, gives us X squared is equal to negative one. Now we have to take the square root of negative one. And understanding that we can't take sort of a negative number, we can say that we do not have any horizontal as some toads or they do not exist. And then for part C here, finding these horizontal assam totes, we know that we just need to take the limit as X approaches infinity of our function and then we really are very interested in our dominant terms here. So we can get rid of the one on the bottom and we're just gonna take this limit as X is approaching infinity of X divided by X squared, which you can see is going to simplify. We'll be able to get rid of these exes right here and we'll be left with just one over X. So if we were to plug a really large number and for ex you could see that would be approaching zero really, really quickly. So we can say here that our horizontal asked until it occurs when y is equal to zero right now, we can go ahead and start to try to graph this at least. So are Y equals zero? That's going to exist just right here along our horizontal axis. That's going to be that as into it. And again, we don't have any vertical ascent oats. Now, just looking at the information we have right here, it's not giving us a whole lot as for where the curves actually take place. So go ahead, plug it into your graphing calculator to get a better idea of what it looks like and what you'll see pretty quickly is that it does follow this general outline that we made for ourselves, right? It is going to cross through the intercept and it's going to follow those asthma. Tote rules. So this function looks something like this

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