How to find the slant asymptote of f(x) = (2x^2 +3) / (x-1) ??

1 Answer
Aug 1, 2018

Essentially by division...

Explanation:

Given:

f(x) = (2x^2+3)/(x-1)

we can divide to get a polynomial quotient and rational remainder as follows:

(2x^2+3)/(x-1) = (2x^2-2x+2x-2+5)/(x-1)

color(white)((2x^2+3)/(x-1)) = (2x(x-1)+2(x-1)+5)/(x-1)

color(white)((2x^2+3)/(x-1)) = 2x+2+5/(x-1)

Then note that:

lim_(x->oo) 5/(x-1) = 0

lim_(x->-oo) 5/(x-1) = 0

So 2x+2 is the slant asymptote of f(x) = (2x^2+3)/(x-1)

graph{(y-(2x^2+3)/(x-1))(y-2x-2) = 0 [-42.17, 37.83, -16.48, 23.52]}