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(a) Find the $x$ -intercept(s); (b) Find the vertical asymptotes; (c) Find the horizontal asymptotes. (d) Sketch the graph.$$f(x)=4 x^{2} /\left(x^{2}+4\right)$$

(a) $x=0$(b) none(c) $y=4$(d)

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 4

Limits at Infinity, Infinite Limits and Asymptotes

Derivatives

Campbell University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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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Graph the function$$f(…

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(a) Determine the $x$ -int…

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in this problem, we are given the function after the X is equal to four, X squared divided by X squared plus four. And we want to find our X intercept and as I'm told, so that we can sketch a graph of it. So part a here finding this X intercept. We're just going to take our entire function for X squared divided by X squared plus four and set it equal to zero. So we can then solve for X. Multiplying both sides by that denominator leaves us with four. X squared is equal to zero, dividing by four on both sides. We have X squared is still equal to zero and obviously the square root of zero is just zero. So we have an X intercept one, X is equal to zero. Go ahead and plot that real quick. I'll sit right in here and for part B we want to find our vertical ascent toe, in which case, what's important to us is our denominator. So X squared plus four, you can just set that equal to zero as well and solve for X again. So we have X squared is to a negative four. And we know we can't take the square root of a negative number, meaning that we don't actually have any vertical asthma totes, so we don't have to worry so much about that one. And for part C finding these horizontal assam totes, we have to take the limit of this function as X approaches infinity. And when taking this limit as we're approaching infinity, we can really just pay attention to these dominant terms of being for X squared and X squared. So we can drop that for on the bottom because ultimately that doesn't make much of a difference when we're approaching infinity. So here, you can see that some of this is going to cancel out right? Are x squares cancel out? And all of a sudden we're just left with four, meaning that our horizontal assam tote occurs, one y Is equal to four. You can go ahead and draw that one. Why is equal to four? We have this vertical ascent. Oh, or this horizontal as into it? Sorry, That sits up top here at Y equals four, meaning we have at least somewhat of an outline That we can follow, but we don't exactly know necessarily the shape of the curve or where it's turning points are. So go ahead and plug the function into your graphing calculator. And then I'll just give you a better idea of what it actually looks like. And then you can see pretty quickly. Also by checking at work that what we found is correct. We do have this horizontal as until that follows why equals for and then it drops down to zero and then it comes back up from zero and again follows that horizontal assam tone. It looks something like this, so you can see, we did follow the asthma told that we found. You can see there's obviously no vertical ascent over here, and it does have the X intercept at X is equal to zero.

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