Question

How do you find the vertical, horizontal and slant asymptotes of: `(3x-2) / (x+1)`?

Answer 1

vertical asymptote x = -1
horizontal asymptote y = 3

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 + 1 = 0 → x = -1 , is the asymptote

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

divide terms on numerator/denominator by x

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

as `x to +- oo , y to (3-0)/(1+0)`

`rArr y = 3 " is the asymptote "`

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