Question

How do I find the asymptotes of `y=7/(3x-4)-1/9`?

Answer 1

vertical asymptote `x=4/3`
horizontal asymptote `y=-1/9`

Explanation:

The denominator of y cannot be zero as this is 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: 3x - 4 = 0 `rArrx=4/3" is the asymptote"`

Horizontal asymptotes occur as

`lim_(xto+-oo),ytoc" (a constant)"`

divide terms on numerator/denominator by x

`(7/x)/((3x)/x-4/x)-1/9=(7/x)/(3-4/x)-1/9`

as `xto+-oo,yto0/(3-0)-1/9`

`rArry=-1/9" is the asymptote"`
graph{(7)/(3x-4)-1/9 [-10, 10, -5, 5]}