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

(a) $x=0$(b) $x=1$(c) $y=2$(d)

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 4

Limits at Infinity, Infinite Limits and Asymptotes

Derivatives

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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.

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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 $y=f(x)$<…

this question is walking us through the idea of intercepts asking totes and limits, essentially forgiven. The function F of X is equal to two, X divided by x minus one and part eight. We want to find our x intercept which most of us have probably done before and all we have to do is set our function equal to zero and solve for X. So you can see we can divide both sides by x minus one to start to simplify. So that gives us two, X is equal to zero and then divide both sides by two. So that we can get to X and you can see that X is still equal to zero. And that gives us our intercept. Let's go ahead and plot that on our graph real quick. So at X equals zero that sits right here at the origin. We know that we're going to have an intercept occur right there when we are graphing and then part B. We are working with vertical ascent oats and what's important with vertical ascent oats is mainly our denominator. We can just take that denominator so x minus one. Set that equal to zero and solve for x. So x minus one equals zero. Let's add one to both sides and we have X is equal to one for our vertical ass until let's go ahead and add that to our graph as well. So when X is equal to one, we have a line that sits in here. I'm just gonna give us these dash lines and then part C. We want to find our horizontal as santos and with horizontal asking told we want to work with these dominant terms in both the numerator and denominator and denominators. That's going to give us two X and an X in the denominator. We can forget about that negative one. It doesn't really matter when it comes to finding the horizontal Assen totes. We really just need to simplify this now so you can see that our exes are going to cancel out and we're left with two. So this is telling us that why is equal to two is our horizontal ass until let's go ahead and put that on the graph as well. So why equals two is going to sit right in here. So now that we have these lines for these dash lines, harassment, oats and our X equals zero for our intercept. We can start to graph this a little bit. So we're going to have a graph on this left hand side that's going to pass through that intercept. But it's not quite going to touch any of these Osama toads. And then on top we have another one that looks like this now the general shape of it. We haven't gotten over yet. How we understand quite these turning points. So I went ahead and I just and put it this function into my graphing calculator to get an idea visually of it. But you can see that it does pass through X equals zero for an intercept, and it does have these assume toads at X equals one and Y equals two.

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