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

(a) $x=0$(b) $x=\pm 1$(c) $y=0$$(d)$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 4

Limits at Infinity, Infinite Limits and Asymptotes

Derivatives

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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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For the graph of $y=f(x)$<…

given the function F of x is equal to X divided by x squared minus one. We want to find the information of its intercepts and ask them to vote so that we can sketch a graph of it. So part A. We want to find the X intercept, which we can do just by taking the entire function and setting it equal to zero. So we can then solve for X can see pretty quickly, we'll just divide both sides by that denominator and we'll be left with X is equal to zero. So you can go ahead and plot that on your axes over here. Part B We want these vertical assam totes, which means we need to pay attention really only to our denominator, X squared minus one. Set that equal to zero. And we can just go ahead and solve for X. Here leaves us with X squared is equal to a positive one, and then we'll take the square root of both sides, which will give us X is equal to a plus or minus right one now, part C. We want to find our horizontal assam. Totes, in which case we need to take the limit as X approaches infinity of our entire function, so X divided by x squared minus one. And we're taking this limit as it's approaching infinity, we can really just pay attention to our dominant terms. Mean we don't have to pay much attention to that negative one in the denominator because in the whole scheme of things minus one won't make a difference if we are approaching infinity. Right? So we can go ahead and drop that and leave ourselves with just X divided by back squared. Simplifying this. Obviously we see that a couple of exes are going to cancel out, which will leave us with one divided by X. So if we were to plug any really large number in for X here, you can see that one divided by that large number is going to be zero, meaning that our horizontal intercept will be when y is equal to zero. Let's go ahead and add these as I'm totes to our graph here. So when y is equal to zero, right, are horizontal, last until that's gonna sit right along this axes. And then for our vertical ascent, toads X equals plus or minus one. So that's probably going to sit somewhere somewhere in here. And if we go ahead and sketch this graph, I think it's helpful to plug it into your graphing calculator. So you can get a better idea of where the turns actually take place because just looking at it right here, you may not be able to tell. So doing that, you can then better sketch it and you can see that the graph does indeed follow these outlines that we gave ourselves right. The X intercept when X is equal to zero and are vertical and horizontal as um toasts. The graph does lie somewhere within all of that.

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