What are the asymptote(s) and hole(s), if any, of f(x)= (x(x-3))/(x^2-9x+3)?

1 Answer
Aug 10, 2018

No hole, Vertical asymptotes : x~~ 0.35 and x~~ 8.65,
Horizontal asymptote: y=1

Explanation:

f(x)= (x(x-3))/(x^2-9 x+3) , since no factor in numerator and

denominator is being cancelled , so there is no hole.

Vertical asymptote occur when denominator is zero.

:. x^2-9 x+3=0 ; a=1 ,b= -9 , c= 3 ; [ax^2+bx+c]

Discriminant D= b^2-4 a c = 81- 12 =69

Quadratic formula: x= (-b+-sqrt D)/(2 a) or

x= (9+-sqrt 69)/2 or x = 4.5+- sqrt 69/2

x~~ 8.65 , x ~~ 0.35 , therefore, vertical asymptotes

are x= 0.35 and x= 8.65

Horizontal asymptote: y= (x^2-3x)/(x^2-9x+3) or

y= (1-3/x)/(1-9/x+3/x^2)

lim(x-> oo) , y= (1-0)/(1-0+0)=1

:. lim(x->+-oo) ,y->1 , hence, horizontal asymptote is

at y=1

graph{(x(x-3))/(x^2-9 x+3) [-40, 40, -20, 20]}[Ans]