Question

Graph the following equation


`f(x) = ((2x+3)(x-6))/((x+2)(x-1))`

If you can. Also, what is the Oblique Asymptote.

Answer 1

Graphic attached

Explanation:

The oblique asymptote of the function has the shape of a line of the form

`y = mx + b`

We need to find the slope m and the intercept b.

The oblique asymptote is found by these two limits:

`m = \lim_(x \to \infty)(f(x))/(x)\\\\b = \lim_(x \to \infty)[f(x) - mx]`

If
`f(x) = ((2x+3)(x-6))/((x+2)(x-1))` then:

`m = \lim_(x \to \infty) (((2x+3)(x-6))/((x+2)(x-1)))/(x)\\\\m = \lim_(x \to \infty) (2x^2-9x-18)/(x^3 +x^2 -2x)\\\\m = 0`

The slope is 0. Therefore the function has no oblique asymptote.

Horizontal asymptote:

`y = 2`

Vertical asymptote

`x = -2\\x = 1`