Question

How do you find any asymptotes of `f(x)=x/(x-5)`?

Answer 1

VA: `x = 5`
HA: `y=1`

Explanation:

(VA) Vertical Asymptote: Set the denominator equal to zero:

`x-5 = 0`

`x = 5`

(HA) Horizontal Asymptote: Divide the coefficients of the x values:

`(1x)/(1x)=1`

`y=1`

Answer 2

`"vertical asymptote at "x=5`
`"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-5=0rArrx=5" 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)/(x/x-5/x)=1/(1-5/x)`

`"as "xto+-oo,f(x)to1/(1-0)`

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