http://www.ck12.org Chapter 2. Polynomials and Rational Functions
f(− 2. 001 ) =(−)·(+)(+)··((−−))·(+)=−f(− 1. 9999 ) =(−)·(+)(+)··((+)−)·(+)= +f( 0. 9999 ) =(+)·(+)(+)··((+)−)·(+)=−f( 1. 0001 ) =(+)·(+)(+)··(+)(+)·(+)= +A sketch of the behavior of this function near the asymptotes is:
Example C
Create a function with two vertical asymptotes at 3 and -2 such that the function approaches positive infinity from
both directions at both vertical asymptotes.
Solution: In Example A there was a function that approached negative infinity from both sides of the asymp-
tote. This occurred because the term was squared in the denominator.
f(x) =(x− 3 ) (^21) (x+ 2 ) 2
Concept Problem Revisited
In order to truly answer the question about how close the numbers need to be, calculus should be used. For the
purposes of PreCalculus, the testing number should be closer to the vertical asymptote than any other number in the
problem. If the vertical asymptote occurs at 3 and 3.01 is in the problem elsewhere, do not choose 3.1 as a sign test
number.