Question

How do you find the Vertical, Horizontal, and Oblique Asymptote given `f(x)=((3x-2)(x+5))/((2x-1)(x+6))`?

Answer 1

vertical asymptotes x = -6 , x`=1/2`
horizontal asymptote `y=3/2`

Explanation:

For this rational function the denominator cannot be zero. 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 values that x cannot be and if the numerator is also non-zero for these values of x then they must be vertical asymptotes.

solve : (2x-1)(x +6 ) =0 `rArrx=-6,x=1/2`

`rArrx=-6" and " x=1/2" are the asymptotes"`

Horizontal asymptotes occur as

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

Now `f(x)=((3x-2)(x+5))/((2x-1)(x+6))=(3x^2+13x-10)/(2x^2+11x-6)`

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

`((3x^2)/x^2+(13x)/x^2-10/x^2)/((2x^2)/x^2+(11x)/x^2-6/x^2)=(3+13/x-10/x^2)/(2+11/x-6/x^2)`

as `xto+-oo,f(x)to(3+0-0)/(2+0-0)`

`rArry=3/2" is the asymptote"`

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (both of degree 2 ) Hence there are no oblique asymptotes.
graph{((3x-2)(x+5))/((2x-1)(x+6) [-10, 10, -5, 5]}