Question

How do you find the asymptotes for `f(x) = ((3x-2)(x+5))/((2x-1)(x+6))`?

Answer 1

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

Explanation:

The denominator of f(x) cannot be zero as this would make f(x) 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: `(2x-1)(x+6)=0rArrx=-6 , x=1/2`

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

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

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`

`f(x)=((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"`
graph{((3x-2)(x+5))/((2x-1)(x+6)) [-10, 10, -5, 5]}