Question

How do you find the Vertical, Horizontal, and Oblique Asymptote given `y = (4 x + 6)/(x - 1)`?

Answer 1

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

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-1=0rArrx=1" is the asymptote"`

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x

`y=((4x)/x+6/x)/(x/x-1/x)=(4+6/x)/(1-1/x)`

as `xto+-oo,yto(4+0)/(1-0)`

`rArry=4" is the asymptote"`

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (both of degree 1) Hence there are no oblique asymptotes.
graph{(4x+6)/(x-1) [-20, 20, -10, 10]}