How do you find the vertical, horizontal and slant asymptotes of: #y=(8x-48)/(x^2-13x+42)#?

1 Answer
Jul 1, 2016

vertical asymptote x = 7
horizontal asymptote y = 0


The first step here is to factorise and simplify y.


The denominator of this rational function cannot be zero as this would lead to division by zero which is undefined.By setting the denominator equal to zero and solving for x we can find the value that x cannot be and if the numerator is also non-zero for this value of x then it must be a vertical asymptote.

solve : x - 7 = 0 → x = 7 is the asymptote

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x


as #xto+-oo.yto0/(1-0)#

#rArry=0" is the asymptote"#

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (numerator-degree 0 ,denominator-degree 1 )Hence there are no slant asymptotes.
graph{8/(x-7) [-20, 20, -10, 10]}