The roots of a Rational function are the roots of its numerator. We can use POLY to find these.
The vertical asymptotes are found by finding the roots of the denominator (again we can use POLY for this). If r is a root of the denominator then the line x = r is a vertical asymptote.
The horizontal asymptotes or slant asymptotes can be found by dividing the
numerator by the denominator to find the quotient polynomial. If the quotient
polynomial is the constant c, then y = c is a horizontal
asymptote. If the quotient polynomial is linear, then it defines a slant
asymptote. These can also be recognized by looking at the graph.
Example 1: Find the roots, and asymptotes of the rational function
f(x) = 
To find the roots of f(x) we just find the roots of 4x2 - 9
Using POLY (or factoring) we find the roots:
x1=3
x2= -3
To find the vertical asymptotes of f(x) we just find the roots of
2x2 + 1
Using POLY we find the roots are both complex. No real roots
then means there are no vertical asymptotes.
Graphing f(x) = in the standard window we see
The horizontal asymptote at y = 2 is clear. You can use
TRACE or view the
TABLE values of the function at large values of x to verify this.