1. Find all vertical asymptotes, horizontal asymptotes, slant asymptotes, and holes
in the graph of the function. Then use a graphing utility to verify your result.
(If an answer does not exist, enter DNE.)
Vertical asymptote x=_________
Horizontal asymptote y=__________
slant asymptote y=___________
hole (x, y) = (___, ___)
Solution
Vertical Asymptotes:
1. Factor the numerator and denominator and reduce the rational expression
(x - 2) (x + 2) (2 x + 1)
------------------------------
(x - 1) (x - 2)
(x + 2) (2 x + 1)
------------------------------
(x - 1)
The vertical asymptote is a vertical line with the equation x = 1.
Hole is at (2, f(2)) by virtue of dividing our the x-2 factor

2. A graph can have a (one) slant asymptote or a (one) horizontal asymptote. Since the degree of
the numerator is exactly 1 greater than the degree of the denominator, the rational expression has
a slant asymptote found by dividing the numerator by the denominator
The slant asymptote is y = 2x + 7