Question

How do you identify all asymptotes for `f(x)=(x+1)/(x-3)`?

Answer 1

`"vertical asymptote at " x=3`
`"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-3=0rArrx=3" 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+1/x)/(x/x-3/x)=(1+1/x)/(1-3/x)`

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

`rArry=1" is the asymptote"`
graph{(x+1)/(x-3) [-11.25, 11.25, -5.625, 5.625]}