Find all the vertical and horizontal asymptotes xintercept a

Find all the vertical and horizontal asymptotes, x-intercept and y-intercept f(x) = 12x^2 - 8x - 4/x^4 + 4x +12

Solution

f(x)=[12x² -8x-4]/(x⁴+4x+12)

x intercept => y =0

[12x² -8x-4]/(x⁴+4x+12) =0

12x² -8x-4=0

3x² -2x-1=0

3x² -3x+x-1=0

(3x+1)(x-1)=0

x =-1/3 , x =1

(x,y)=(-1/3,0),(1,0)

for y intercept x =0

y=[0 -0-4]/(0+0+12)

y =-1/3

(x,y)=(0,-1/3)

for vertical asymptote denominator =0

(x⁴+4x+12) =0

no vertical asymptote

horizontal asymptote :

y =lim_x->[12x² -8x-4]/(x⁴+4x+12)

y=lim_x->x²[12 -8/x -4/x²]/(x⁴(1+ 4/x³ +12/x⁴))

y=lim_x->[12 -8/x -4/x²]/(x²(1+ 4/x³ +12/x⁴))

y=[12 -0-0]/((1+ 0+0))

y=0