Question

What are the vertical and horizontal asymptotes of `f(x)=5/((x+1)(x-3))`?

Answer 1

`"vertical asymptotes at "x=-1" and "x=3`
`"horizontal asymptote at "y=0`

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

`rArrx=-1" and "x=3" 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)=(5/x^2)/(x^2/x^2-(2x)/x^2-3/x^2)=(5/x^2)/(1-2/x-3/x^2)`

`"as "xto+-oo,f(x)to0/(1-0-0)`

`rArry=0" is the asymptote"`
graph{5/((x+1)(x-3)) [-10, 10, -5, 5]}