Question

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

Answer 1

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`