Question

How do you find the vertical, horizontal or slant asymptotes for `f(x) = (x-3)/(x-2)`?

Answer 1

vertical asymptote x = 2
horizontal asymptote y = 1

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 : x - 2 = 0 → x = 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-3/x)/(x/x-2/x)=(1-3/x)/(1-2/x)`

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

`rArry=1" is the asymptote"`

Slant 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 slant asymptotes.
graph{(x-3)/(x-2) [-10, 10, -5, 5]}