Finding the horizontal and vertical asymptotes of each curve

[winterrose]

Hello I am stuck on this question:Finding the horizontal and vertical asymptotes of each curve.

My function is y= 1+x^4/x²-x⁴


So what I first was find the x intercepts to get my v.a. (vert asymp.) by setting the denominator to zero.

• 0=x²-x⁴
• 0=x²(1-x²)
• 0=x² and 0=1-x²
• x=0, and x= +1 and -1

That would mean that my domain would be: All real numbers except at x=-1,0,1.

Then I got a little confused but from here to get my horizontal asympotes I need to take the limit of all them right? So I would take the limit as x approaches -1 from the left and from the right. And as x approaches 0 from both directions, and the same for 1. Is this correct? Maybe my algebra is off?

Any help would be greatly appreciated.

 

[DrPhil]

Hello I am stuck on this question:Finding the horizontal and vertical asymptotes of each curve.

My function is y= (1+x⁴)/(x²-x⁴)


So what I first was find the x intercepts to get my v.a. (vert asymp.) by setting the denominator to zero.

• 0=x²-x⁴
• 0=x²(1-x²)
• 0=x² and 0=1-x²
• x=0, and x= +1 and -1

That would mean that my domain would be: All real numbers except at x=-1,0,1.

Then I got a little confused but from here to get my horizontal asympotes I need to take the limit of all them right? So I would take the limit as x approaches -1 from the left and from the right. And as x approaches 0 from both directions, and the same for 1. Is this correct? Maybe my algebra is off?

Any help would be greatly appreciated.

Asymptotic behavior is how the complete function behaves as x increases (or decreases) without limit. For a rational expression, keep the highest power term in the numerator, and the highest power term in the denominator

\(\displaystyle \displaystyle \lim_{x \to \infty}\dfrac{1 + x^4}{x^2 - x^4} = \dfrac{x^4}{-x^4} = -1 \)

for either + or - x. Note that the asymptote will be horizontal if (and only if) the highest power in the numerator is the same as in the denominator.

 

[winterrose]

Thanks for you comment Dr.Phil. Why not take the limit of the x-intercepts?

 

[lookagain]

Note that the asymptote will be horizontal if (and only if)the highest power in the numerator is the same as in the denominator.

DrPhil, that is not true. \(\displaystyle \ \ \) If we concern ourselves with rational functions (quotients of polynomials),

there will be a horizontal asymptote if and only if the degree of the numerator is less than or equal to the degree

of the denominator. \(\displaystyle \ \ \) In the case where the degree of the numerator is strictly less than the degree

of the denominator, the equation of the horizontal asymptote is \(\displaystyle \ y = 0.\)