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

(a) $x=0$(b) none(c) $y=\pm 2$(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

Baylor University

University of Nottingham

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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given the function two X divided by the square root of X squared plus one. We want to find its X intercepts. And as some totes in order to draw a sketch of it. So starting with part A finding its X intercepts. We know we just need to set the function equal to zero and solve for X. So doing this, we're gonna multiply that right hand side by what we have in the denominator here, which still just leaves us with two. X is equal to zero. And then obviously dividing zero by two is going to give us X is equal to zero for our intercept. Go ahead and plot that right here at the origin. And now for part B to find these vertical assam totes, we like to use our denominators. So the square root of X squared plus one, set that equal to zero. And then we can again just solve for X so square to both sides. Obviously that still leaves us with X squared plus one equal to zero. We then have X squared is equal to negative one. And we can't take skirt of a negative number, meaning that we do not have any vertical Assen totes so it does not exist or none. All right. And then for part C finding these horizontal ass and tell us what we have to do is take the limit of our function as X is approaching infinity. And taking this limit, we want to look at our dominant terms both in the numerator and denominator. This is going to reduce down to taking the limit of just two X and the new writer. It's the only terms that can be the only one to use. And then in the denominator we see we have X squared but really we're taking the square root of it. So the square root of X squared is just X. You can see here that are exes will cancel and we're just left with two, meaning that our horizontal absent oat occurs when y is equal to two. And actually, because we were working with the square root right here, we don't want to forget about that, it's actually gonna be Y is equal to plus or minus two. So we can go ahead and start to sketch this. So when y is equal to plus and minus two, we'll give ourselves these ascent to its and then to sketch graph, obviously it's going to cross through X equals zero and stay within these as um toasts. So if you want, you can go ahead and plug it into a graphing calculator to get a better a better visual picture of it. But you'll see pretty quickly that it absolutely follows what we just discovered about the ASM toads and the intercept. And it looks something like that.

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