Question

How do you find the Vertical, Horizontal, and Oblique Asymptote given `y= (x + 1) / (2x - 4)`?

Answer 1

vertical asymptote x = 2
horizontal asymptote `y=1/2`

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 :`2x-4=0rArrx=2" is the asymptote"`

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x

`(x/x+1/x)/((2x)/x-4/x)=(1+1/x)/(2-4/x)`

as `xto+-oo,yto(1+0)/(2-0)`

`rArry=1/2" 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{(x+1)/(2x-4) [-10, 10, -5, 5]}