What are the asymptotes and removable discontinuities, if any, of f(x)=( 2x^2 + 3x-2)/(x-2)?

1 Answer
Apr 13, 2017

Vertical asymptote at x = 2
Slant asymptote: y = 2x+7
No removable discontinuities

Explanation:

For rational functions f(x) = (N(x))/(D(x))= (a_nx^n + ...)/(b_mx^m+...)

Factor the numerator to see if there are any removable discontinuities (holes):

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

Nothing cancels, so No removable discontinuities

Vertical asymptotes found when D(x) = 0:
Vertical asymptote at x = 2

When m + 1 = n you have a slant asymptote:
m = 1, n= 2" m + 1 = n = 2# so we have a slant asymptote.

Slant asymptotes are found by polynomial long division or synthetic division:

f(x) = (2x^2 + 3x - 2)/(x - 2) = 2x + 7 +12/(x-2)

The slant asymptote: y = 2x + 7