Question

How do you find vertical, horizontal and oblique asymptotes for `y= x/((x+3)(x-4)`?

Answer 1

`"vertical asymptotes at " x=-3" and " x=4`

`"horizontal asymptote at " y=0`

Explanation:

The denominator of y cannot be zero as this would make y undefined. Equating the denominator yo zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes.

`"solve " (x+3)(x-4)=0`

`rArrx=-3" and " x=4" are the asymptotes"`

`"Horizontal asymptotes occur as"`

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

`"divide terms on numerator/denominator by the highest power"`
`"of x, that is " x^2`

`y=(x/x^2)/(x^2/x^2-x/x^2-12/x^2)=(1/x)/(1-1/x-12/x^2)`

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

`rArry=0" is the asymptote"`
graph{x/((x+3)(x-4)) [-10, 10, -5, 5]}