Question

How do you find vertical, horizontal and oblique asymptotes for `(3x+5)/(x-6)`?

Answer 1

vertical asymptote x = 6
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 - 6 = 0 → x = 6 is the asymptote

Horizontal asymptotes occur as `lim_(xto+-oo) , y to 0`

divide terms on numerator/denominator by x

`rArr((3x)/x+5/x)/(x/x-6/x)=(3+5/x)/(1-6/x)`

as `xto+-oo ,yto(3+0)/(1-0)`

`rArry=3" is the asymptote"`

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (both degree 1).Hence there are no oblique asymptotes.
graph{(3x+5)/(x-6) [-20, 20, -10, 10]}