Question

How do you identify all asymptotes or holes for `f(x)=(x^2-1)/(-3x^2+12)`?

Answer 1

vertical asymptotes at x = ± 2
horizontal asymptote at `y=-1/3`

Explanation:

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+12=0rArr3x^2=12rArrx^2=4rArrx=+-2`

`rArrx=-2" and " x=2" 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^2/x^2-1/x^2)/((-3x^2)/x^2+12/x^2)=(1-1/x^2)/(-3+12/x^2)`

as `xto+-oo,f(x)to(1-0)/(-3+0)`

`rArry=-1/3" is the asymptote"`

Holes occur when there are duplicate factors on the numerator/denominator. There are no duplicate factors here hence there are no holes.
graph{(x^2-1)/(-3x^2+12) [-10, 10, -5, 5]}