Problem

A function $ f $ is a ratio of quadratic function…

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Problem 58 Medium Difficulty

Find a formula for a function that has vertical asymptotes $ x = 1 $ and $ x = 3 $ and horizontal asymptote $ y = 1 $.

Answer

$f(x)=\frac{x^{2}}{x^{2}-4 x+3}$

Topics

Limits

Derivatives

Book Cover for Calculus: Early Tra…

Calculus: Early Transcendentals

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Discussion

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Video Transcript

this bomb number fifty eight of this tour Calculus eighth edition section two point six Find a formula for a function that has vertical. Hasn't it's X equals one in X equals three and horizontal as do that. Why equals to one So we can begin first with the vertical aspen, toots. An easy way, Tio No, exactly Where to put a bird qassem toe is to make sure that the denominator equal zero when wherever you want to get rid of glass which had to be here we wanted to be at X equals to one and it x equals three. So you have two terms in the denominator experience one an ex ministry. This guarantees that our function undefined it those families meaning that there's at this continuity. And as long as we don't have anything in the numerator that removes this concentrating, then we can be sure that those vertical isn't it remained with classmates. Horizontal has to do with the limit as expert tunes. Ah infinity! And we want this to be equal two one down. Let's work with what we have right now and the denominator we haven't x squared. Every oil is out. X squared mass for X extreme. And then we're just trying to come up with something. The numerator. It makes us limit goto one. As long as we have a function the same degree that will guarantee that our limit exists and is equal to the coefficient. The ratio that coefficients here. And since then, coefficient of our aunt Denominator of the highest order term is one. We're going to leave this this one and this is this makes our limit. It will be equal to one. If we complete this limited of equal to one, which means our workers, our class will be at one, which means that our numerator should be X squared. And then over on we found a function that has two requested at X equals one X equals three and definitely has a horizontal husband. And why close to one

Daniel J.

Topics

Limits

Derivatives

Book Cover for Calculus: Early Tra…

Calculus: Early Transcendentals

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Top Calculus 1 / AB Educators

Grace H.

Numerade Educator

Kristen K.

University of Michigan - Ann Arbor

Michael J.

Idaho State University

Joseph L.

Boston College