Give the equations of any vertical, horizontal, or oblique asympt...

Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=(x^2+1)/(x^2+9)

Accepted Answer

Hey, everyone in this problem for the following function, we're asked to write the equations for the vertical, horizontal or oblique asymptotes. We're given the function F of X is equal to X squared plus 16 divided by X squared plus 25. We have four options for answers. Option A, the horizontal axent is at Y is equal to one. Option B we have a vertical Asymptote at Y is equal to one option C we have a horizontal Asymptote at Y is equal to negative one. And option D we have a vertical Asymptote at X is equal to negative 25 a horizontal Asymptote at Y is equal to one. So let's start rewriting our function. We have F of X is equal to X squared plus 16 divided by X squared plus 25. And we're gonna start with our horizontal acid. Now, when we're looking for horizontal asymptotes, we wanna look at the degree of the numerator and the degree of the denominator in our function. Now, the degree of the numerator in this case is going to be equal to two. OK. That's the highest exponent that we have. In this case is two X squared and this is actually equal to the degree of the denominator. OK. The degree of the denominator is also two, the highest exponent term is X squared. OK. So we have the degree of the numerator is equal to the degree of the denominator. So we have a horizontal Asymptote and the horizontal Asymptote is going to occur at Y is equal to the ratio of the leading coefficients. So we're gonna take the leading coefficient of the numerator and divided by the leading coefficient of the denominator. And you'll notice that I've written the horizontal asom tote as Y equals OK. The horizontal Asymptote is a horizontal line. If you imagine drawing a horizontal line on your graph that has an equation Y equals OK. So your horizontal Asymptote will always be Y equals not X equals all right. So our leading coefficient is the coefficient corresponding with that highest exponent term. So in the numerator, our highest exponent term is X squared. The coefficient is just one. So we get one then in the denominator again, our highest exponent term is X squared and it has a coefficient of one as well. So our horizontal Asymptote is going to be at Y is equal to one divided by one which is just equal to one. OK. So we figured out our horizontal atom tote now because we have a horizontal AOM tote. That means we won't have an oblique as to OK. We can't have both because the degree of the numerator is the same as the denominator. We have a horizontal Asom tope, we do not have an oblique Asom tobe. So all we're left to do is look at the vertical Asom tube and the vertical Asymptote is going to occur where the denominator is equal to zero. And when I say the denominator is equal to zero, what I mean is after simplifying. OK. So we want to simplify, we wanna cancel any terms that can be canceled. And then we want to see where it is equal to zero. So looking at our function, we have X squared plus 16 in the numerator, we can't factor that any further. We can't simplify same thing in the denominator. We have X squared plus 25. We can't factor that. We can't simplify. So we're gonna look just where that denominator is equal to zero. We have X squared plus 25 is equal to zero. OK. If we try to solve, we get that X squared is equal to negative 25 but X squared is always gonna be a positive value. OK. So X squared is never going to be equal to negative 25. So there are no real solutions. OK. That means that there's nowhere where the denominator is equal to zero. And so we have no vertical atom to all right. So we found that we have a horizontal Asymptote and that's it. And that horizontal Asymptote occurs at Y is equal to one. This corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

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Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4. ƒ(x)=(x^2+1)/(x^2+9)