Question

How do you find the asymptotes for `(3x-12)/( 4x-2)`?

Answer 1

Horizontal asymptotes of a rational function occurs when the function in the denominator becomes zero.
In this case the function in denominator is `4x-12`

For horizontal asymptotes `4x-12=0` `implies x-3=0 implies x=3`

Hence horizontal asymptote is `3`

Vertical asymptotes accurs when the degree of numerator and denominator is equal. In this case both numerator and denominator have a degree `1` i.e, the degree is equal. The asymptote is found out by the ratio of leading coefficients.
In this case the leading coefficients of numerator and denominator are `3` and `4` respectively.
`implies` vertical asymptote`=3/4`

Answer 2

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

Explanation:

Vertical asymptotes can be found when the denominator of

the rational function is zero.

This will be when : 4x - 2 =0 hence 4x = 2 so x `= 1/2`

[ Horizontal asymptotes can be found when the degree of the

numerator and the degree of the denominator are equal ]

In this question they are both of degree 1 and so equal.

The asymptote can be found by taking the ratio of leading

coefficients hence y = `3/4`

graph{(3x-12)/(4x - 2) [-22.5, 22.5, -11.25, 11.25]}