Question

Graph the following equation

Answer 1

Answer:

Graphic attached

Step-by-step 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:

$m = \lim_{x \to \infty}\frac{f(x)}{x}\\\\b = \lim_{x \to \infty}[f(x) - mx]$

If $f(x) = \frac{(2x+3)(x-6)}{(x+2)(x-1)}$ then:

$m = \lim_{x \to \infty} \frac{\frac{(2x+3)(x-6)}{(x+2)(x-1)}}{x}\\\\m = \lim_{x \to \infty} \frac{2x^2-9x-18}{x^3 +x^2 -2x}\\\\m = 0$

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

Horizontal asymptote:

$y = 2$

Vertical asymptote

$x = -2\\x = 1$