Find the vertical horizontal and oblique asymptotes if any o

Find the vertical, horizontal, and oblique asymptotes, if any, of the rational functions F(x)= 5/x^2 - 1, G(x) = x/(x-1)(x+4), and R(x) = 6x^2 + x+ 12/3x^2 - 5x - 2

Solution

F(x) = 5/(x^2 -1)

Horizontal asymtotes:

x^2 -1 =0 ; x=+1 and x=-1 are two vertical asymtotes

As degree of numerator is less than degree of denominator,so no oblique asymtotes.

G(x) = x/(x-1)(x+1)

Horizontal asymtote: numerator is a lower degree than the denominator, the x-axis (y = 0) is the horizontal asymptote.

Vertical asymtote: (x-1)(x+4) =0 ; x=1 and x=-4 are the vertical asymtotes

Oblique Asymtotes : No oblique asymtote.

R(x) = (6x^2 +x+12)/(3x^2 -5x -2)

Horizontal asymtote: If both polynomials are the same degree, divide the coefficients of the highest degree terms. y = 6x^2/3x^2 =2; y=2

Vertical asymtote : 3x^2 -5x -2 =0

3x^2 -6x + x -2 =0 ; 3x(x -2) +1(x-2) =0

(3x+1) (x-2) =0 ; x= -1/3 ; x=2

A slant (oblique) asymptote occurs when the polynomial in the numerator is a higher degree than the polynomial in the denominator.

No oblique asymtotes exist