Question

How do you find the vertical, horizontal and slant asymptotes of: `y= (3x+5)/(x-6)`?

Answer 1

vertical asymptote at x =6
horizontal asymptote at y = 3

Explanation:

The denominator of y cannot be zero as this would make y undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve : `x-6=0rArrx=6" is the asymptote"`

Horizontal asymptotes occur as

`lim_(xto+-oo),ytoc" ( a constant)"`

divide terms on numerator/denominator by x

`y=((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"`

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{(3x+5)/(x-6) [-20, 20, -10, 10]}