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(a) Find the $x$ -intercept(s); (b) Find the vertical asymptotes; (c) Find the horizontal asymptotes. (d) Sketch the graph.$$f(x)=(1+2 x) /(x-3)$$

(a) $x=-1 / 2$(b) $x=3$(c) $y=2$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 4

Limits at Infinity, Infinite Limits and Asymptotes

Derivatives

Missouri State University

Harvey Mudd College

University of Nottingham

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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For the graph of $y=f(x)$<…

given the function one plus two, X over x minus three. What we want to do is start by finding our X intercepts vertical ascent toads and horizontal asuntos that we can schedule graph of it at the end. So starting with part A here, finding our X intercept, we just need to set our function equal to zero and solve for X. It's going ahead, multiplying both sides by x minus three. We then are left with one plus two, X is equal to zero. Let's subtract one from both sides to give us too accessible to negative one and then divide both sides by two. So we can have X is equal to a negative one half and that's going to be our X intercept, which will sit right in here. Part B We want to find our vertical as until what's important here is to really look at our denominators, we can take x minus three and just set that equal to zero, solve for X and that'll give us that vertical ascent old. So you can see here that X is equal to a positive three. So we can go ahead and sketch that out as well. That sits right in here and then for part C we want the horizontal assume total for this. We really just want to take a look at our dominant terms about the numerator and denominator. So up top, we have two X in the denominator we have X. And let's just simplify this. You can see our excess will cancel out more left with two. So when Y is equal to two, we're going to have a horizontal ass into it. So we can go ahead and sketch that as well. Now, if we go ahead and grab us out, what I think is easiest to do sometimes is to just type this function into my crafting calculator so that I can get a better idea of where the turns are going to actually occur along these, along these lines, where they're actually going to change direction, even though we have this kind of rough outline here. So we can see is that we have a graph that starts somewhere in here. We definitely are going to go through that intercept, like we said, and get really, really close to our ass and totes. And then on top we have one that looks something like that. So if you plug this in your graphing calculator, obviously it's going to look much better than my sketch here. But what's important to note is that our graph is following the as adults, we found and the intercept.

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