Domains and Asymptotes

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Question

Domains and Asymptotes

Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.

y = 2x / (x² − 1)

Accepted Answer

Welcome back, everyone. Find the domain and the asymptotes of the given function Y equals x + 1 divided by x2 minus 4. What we're going to do is begin with the domain of this function because we're given a rational function, we have to recall that the denominator cannot be equal to 0 because division by 0 is undefined, so that x2 minus 4 is not equal to 0. We can perform factorization of the difference of squares to get a factored form of x minus 2 multiplied by X plus 2 and this cannot be equal to 0, which means that Any of our two factors must not be equal to 0, right? So X cannot be equal to -2. And X cannot be equal to 2. So if we write our domain, we can say that. Our domain is all real numbers except from negative or positive 2, which can be written as. A Belongs to the interval from negative infinity up to -2, and from -2 up to 2. And from 2 up to infinity. And that's our first answer to this problem, and now we can continue with the asymptotes. Let's begin with the vertical asymptote, and we have to recall that for a vertical asymptote we want to ensure that our numerator X + 1 is not equal to 0, but our denominator X2 minus 4 is equal to 0. We already know that we have two candidates for the denominator to be equal to 0. Those would be X equals positive or negative 2, and those values do not make the numerator equal to 0, meaning they are both asymptotes, right? We don't have any holes. So we can conclude that the vertical asymptotes are X equals -2, and X equals 2. So those would be our two vertical asymptotes. For the horizontal asymptotes. We essentially want to evaluate the limits. As x approaches positive or negative infinity of the given function, so X plus 1 divided by X2 minus 4. If we simply substitute infinity, we're going to get infinity divided by infinity, which is an indeterminate form. So what we're going to do is simply divide both sides of our fraction by X with the highest power in the denominator, which is X2. So dividing our fraction by X2, we're going to get X divided by X2, which is 1 divided by X in the numerator. Plus 1 divided by x squared. And the numerator again and for the denominator we get 1 minus 4 divided by x 2. All the fractional terms are going to approach 0 because a non-zero number divided by an infinitely large positive or negative number is going to give us a limit of 0. So whether we approach positive or negative infinity, the limit is going to be 0 divided by 1, which is 0. And this gives us our horizontal asymptote in the form of Y equals 0, which is our final answer to this problem. Thank you for watching.

Domains and Asymptotes

Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.

y = 2x / (x² − 1)

Key Concepts

Domain of a Function

Limits and Asymptotes

Rational Functions

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