Question

What are the asymptotes and removable discontinuities, if any, of `f(x)= (x^2+x-12)/(x^2-4)`?

Answer 1

`"vertical asymptotes at "x=+-2`
`"horizontal asymptote at "y=1`

Explanation:

`"factorise numerator/denominator"`

`f(x)=((x+4)(x-3))/((x-2)(x+2))`

`"there are no common factors on numerator/denominator"`
`"hence there are no removable discontinuities"`

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 "(x-2)(x+2)=0`

`rArrx=+-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+x/x^2-12/x^2)/(x^2/x^2-4/x^2)=(1+1/x-12/x^2)/(1-4/x^2)`

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

`rArry=1" is the asymptote"`
graph{(x^2+x-12)/(x^2-4) [-20, 20, -10, 10]}