How do you identify all horizontal and slant asymptote for #f(x)=(3x^2+1)/(x^2+x+9)#?

Question

Elias Ackerman · Accepted Answer

Horizontal: #larr y = 3 rarr#

By actual division,

#f(x) = 3-(3x+26) / (x^2+x+9)#

y = quotient = 3 gives the asymptote.

For x <= -26/3, y>=3#.

On the left side, (-10, 3.04040404..)# is on the curve.

Yet, as #x to -oo, y to 3#, turning back to the asymptote

graph{y(x^2+x+9)-3x^2-1 = 0 [-10, 10, -5.21, 5.21]}

Lincoln Anderson · Answer

horizontal asymptote at y = 3

Horizontal asymptotes occur as #lim_(xto+-oo),f(x)toc" ( a constant)"# divide terms on numerator/denominator by the highest power of x, that is #x^2# #f(x)=((3x^2)/x^2+1/x^2)/(x^2/x^2+x/x^2+9/x^2)=(3+1/x^2)/(1+1/x+9/x^2)# as #xto+-oo,f(x)to(3+0)/(1+0+0)# #rArry=3" is the asymptote"# Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (both of degree 2) Hence there are no slant asymptotes. graph{(3x^2+1)/(x^2+x+9) [-10, 10, -5, 5]}

Benjamin Achtenberg · Answer

undefined To identify horizontal and slant asymptotes for the function $ f(x) = \frac{3x^2 + 1}{x^2 + x + 9} $, follow these steps:

1. **Horizontal Asymptotes**:
   - Compare the degrees of the numerator and the denominator.
   - If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $ y = 0 $.
   - If the degree of the numerator equals the degree of the denominator, divide the leading coefficients to find the horizontal asymptote.
   - If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

2. **Slant Asymptote**:
   - A slant asymptote exists when the degree of the numerator is one more than the degree of the denominator.
   - To find the slant asymptote, perform polynomial long division or synthetic division.
   - Divide the numerator by the denominator to obtain the quotient.
   - The quotient represents the equation of the slant asymptote.

By applying these steps to $ f(x) = \frac{3x^2 + 1}{x^2 + x + 9} $, we can determine the presence and equations of any horizontal and slant asymptotes.