Question

How do you find vertical, horizontal and oblique asymptotes for `(3x-12)/(4x-2)`?

Answer 1

vertical asymptote `x=1/2`
horizontal asymptote `y=3/4`

Explanation:

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation set the denominator equal to zero.

solve : 4x - 2 = 0 → 4x = 2

`rArr x=1/2" is the asymptote "`

Horizontal asymptotes occur as `lim_(x to +- oo) , f(x) to 0`

divide terms on numerator/denominator by x

`((3x)/x-12/x)/((4x)/x-2/x)=(3-12/x)/(4-2/x)`

as `x to +- oo , y to (3-0)/(4-0)`

`rArr y=3/4" is the asymptote "`

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (numerator-degree 1 and denominator-degree 1)hence there are no oblique asymptotes.
graph{(3x-12)/(4x-2) [-10, 10, -5, 5]}