Question

What are the asymptotes and removable discontinuities, if any, of `f(x)=(x-12)/(2x-3)`?

Answer 1

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

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 : 2x - 3 = 0 `rArrx=3/2" is the asymptote"`

Horizontal asymptotes occur as

`lim_(xto+-oo),f(x)toc" (a constant)"`

divide terms on numerator/denominator by x

`(x/x-12/x)/((2x)/x-3/x)=(1-12/x)/(2-3/x)`

as `xto+-oo,f(x)to(1-0)/(2-0)`

`rArry=1/2" is the asymptote"`

There are no removable discontinuities.
graph{(x-12)/(2x-3) [-10, 10, -5, 5]}