Question

How do you find the vertical, horizontal and slant asymptotes of: `f(x)=(2x+4)/(x^2-3x-4)`?

Answer 1

vertical asymptotes x = -1 , x = 4
horizontal asymptote y = 0

Explanation:

The denominator of f(x) cannot be zero as this is undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes.

solve: `x^2-3x-4=0rArr(x-4)(x+1)=0`

`rArrx=-1" and " x=4" are the asymptotes"`

Horizontal asymptotes occur as

`lim_(xto+-oo),f(x)toc" (a constant)"`

divide terms on numerator/denominator by the highest power of x, that is `x^2`

`((2x)/x^2+4/x^2)/(x^2/x^2-(3x)/x^2-4/x^2)=(2/x+4/x^2)/(1-3/x-4/x^2)`

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

`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 1, denominator-degree 2)Hence there are no slant asymptotes.
graph{(2x+4)/(x^2-3x-4) [-10, 10, -5, 5]}