Question

How do you find vertical, horizontal and oblique asymptotes for `((x-1)(x-3))/(x(x-2))`?

Answer 1

vertical asymptotes x = 0 , x = 2
horizontal asymptote y = 1

Explanation:

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

solve: x(x-2) = 0 → x = 0 , x = 2 are the asymptotes

Horizontal asymptotes occur as

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

now `((x-1)(x-3))/(x(x-2))=(x^2-4x+3)/(x^2-2x)`

divide terms on numerator/denominator by `x^2`

`(x^2/x^2-(4x)/x^2+3/x^2)/(x^2/x^2-(2x)/x^2)=(1-4/x+3/x^2)/(1-2/x)`

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

`rArry=1" 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 2 ) Hence there are no oblique asymptotes.
graph{(x^2-4x+3)/(x^2-2x) [-10, 10, -5, 5]}