Question

How do you identify all asymptotes for `f(x)=-(x+2)/(x+4)`?

Answer 1

`"vertical asymptote at " x=-4`
`"horizontal asymptote at " y=-1`

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 value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

`"solve " 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)=-(x/x+2/x)/(x/x+4/x)=-(1+2/x)/(1+4/x)`

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

`rArry=-1" is the asymptote"`
graph{-(x+2)/(x+4) [-10, 10, -5, 5]}