Question

How do you identify all asymptotes for `f(x)=(x-1)/(x-4)`?

Answer 1

vertical asymptote at x = 4
horizontal asymptote at y = 1

Explanation:

The denominator of y cannot be zero as this would make y 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),ytoc" ( a constant)"`

divide terms on numerator/denominator by x

`y=(x/x-1/x)/(x/x-4/x)=(1-1/x)/(1-4/x)`

as `xto+-oo,yto(1-0)/(1-0)`

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