How do you find the vertical, horizontal or slant asymptotes for #(x-7)/(3x^2+17x-6 )#?

Question

Serenity Jones · Accepted Answer

#"vertical asymptotes at "x=-6" and "x=1/3#
#"horizontal asymptote at "y=0# #"let "f(x)=(x-7)/(x^2+17x-6)# The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes. #"solve: "3x^2+17x-6=0rArr(3x-1)(x+6)=0# #rArrx=-6" and "x=1/3" are the asymptotes"# 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)=(x/x^2-7/x^2)/((3x^2)/x^2+(17x)/x^2-6/x^2)=(1/x-7/x^2)/(3+(17)/x-6/x^2)# as #xto+-oo,f(x)to(0-0)/(3+0-0)# #rArry=0" is the asymptote"# Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (numerator-degree 1, denominator-degree 2 ) Hence there are no slant asymptotes. graph{(x-7)/(3x^2+17x-6) [-10, 10, -5, 5]}

Caleb Alexander · Answer

undefined To find the vertical, horizontal, or slant asymptotes for the function $\frac{x-7}{3x^2+17x-6}$, follow these steps:

1. **Vertical Asymptotes:**
   Vertical asymptotes occur where the denominator of the rational function becomes zero, but the numerator does not. To find vertical asymptotes, set the denominator equal to zero and solve for $x$.

$$3x^2 + 17x - 6 = 0$$

You can solve this quadratic equation using the quadratic formula or factoring.

2. **Horizontal Asymptotes:**
   Horizontal asymptotes occur when $x$ approaches positive or negative infinity. To find horizontal asymptotes, compare the degrees of the numerator and denominator of the rational function.

If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
   If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients.
   If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.

3. **Slant Asymptotes (Oblique Asymptotes):**
   Slant asymptotes occur when the degree of the numerator is one more than the degree of the denominator. To find slant asymptotes, perform polynomial long division or use synthetic division to divide the numerator by the denominator. The quotient represents the equation of the slant asymptote.

Perform polynomial long division or synthetic division:
   $$ \frac{x-7}{3x^2+17x-6} = \text{quotient} + \frac{\text{remainder}}{3x^2+17x-6} $$

If the remainder is a non-zero constant, then there is a slant asymptote. The equation of the slant asymptote is the quotient obtained from the division.

By following these steps, you can determine the vertical, horizontal, and slant asymptotes of the given rational function.