Question

How do you identify all asymptotes or holes for `f(x)=x/(-4x^2-16x)`?

Answer 1

`"vertical asymptote at "x=-4`
`"horizontal asymptote at "y=0`

Explanation:

`"simplify f(x) by factorising the denominator"`

`f(x)=cancel(x)/(-4cancel(x)(x+4))=-1/(4(x+4))`

`"the removal of the factor x from the numerator/denominator"`
`"indicates a hole at x = 0"`

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

`"solve "4(x+4)=0rArrx=-4" is the asymptote"`

`"horizontal asymptotes occur as "`

`lim_(xto+-oo),f(x)toc" ( a constant)"`

`"divide terms on numerator/denominator by x"`

`f(x)=-(1/x)/((4x)/x+16/x)=-(1/x)/(4+16/x)`

as `xto+-oo,f(x)to0/(4+0)`

`rArry=0" is the asymptote"`
graph{x/(-4x^2-16x) [-10, 10, -5, 5]}