Question

State the horizontal asymptote of the rational function. f(x) = quantity x squared plus three x minus two divided by quantity x minus two. Please explain as well.

Answer 1

$f(x)=\dfrac{x^2+3x-2}{x-2}\\ D_f=\mathbb{R}\setminus\{2\}\\\\ \displaystyle \lim_{x\to \infty}\dfrac{x^2+3x-2}{x-2}=\lim_{x\to \infty}\dfrac{x\left(x+3-\dfrac{2}{x}\right)}{x\left(1-\dfrac{2}{x}\right)}=\lim_{x\to \infty}\dfrac{x+3-\dfrac{2}{x}}{1-\dfrac{2}{x}}=\dfrac{\infty}{1}=\infty\\ \lim_{x\to -\infty}\dfrac{x^2+3x-2}{x-2}=\lim_{x\to -\infty}\dfrac{x\left(x+3-\dfrac{2}{x}\right)}{x\left(1-\dfrac{2}{x}\right)}=\lim_{x\to -\infty}\dfrac{x+3-\dfrac{2}{x}}{1-\dfrac{2}{x}}=\dfrac{-\infty}{1}=-\infty$

The limits are not numbers, so the asymptote doesn't exist.