Limits of rational functions
Evaluating the limit of a rational function at a point
The limit of a rational function that is defined at the given point
The limit of a rational function that is not defined at the given point
Evaluating the limit of a rational function at a point
A rational function is continuous at every x except for the zeros of the denominator.
Therefore, all real numbers x except for the zeros of the denominator, is the domain of a rational function.
The zeros of the numerator that are in the domain are the x-intercepts or roots of a rational function.
If  x = a is a zero of the numerator and not a zero of the denominator, then  f (a) = 0 or a is the root of the rational function. While if both the numerator and denominator are zero, we get the indeterminate form 0/0.
Factoring the numerator and denominator into irreducible factors allows us to find all of the zeros of the numerator and denominator.
a)  The limit of a rational function that is defined at the given point
 Given a rational function where p (x) and q (x) are polynomials, to find
we first evaluate  p (a) and q (a) by substituting  x = a  into both polynomials then
 if q (a) ¹ 0 then f (x) is continuous at a and the limit is
 Example:  Evaluate the limit
 Solution:  We first factor the numerator and denominator Since q(1/2)  is not 0 then x = 0 and x = 1 are vertical asymptotes, and  y = 1 is the horizontal asymptote, as is shown in the right figure.
b)  The limit of a rational function that is not defined at the given point
At every point that is a zero of the denominator a rational function has either a vertical asymptote or a hole in the graph cased by the indeterminate form 0/0.
The vertical asymptote is called the infinite discontinuity while the hole in the graph is called removable discontinuity since the indeterminate form can be avoided by canceling common factors in the numerator and the denominator.
Thus, a point of discontinuity or hole in the graph exists when a zero of the numerator is matched by a zero of the denominator and the factor occurs to the same degree in the numerator and the denominator.
 Given a rational function then,
 - if  q(a) = 0 and  p(a) is not 0 then one-sided limits are infinite limits.
That means, the rational function has the vertical asymptote at x = a.
- if  q (a) = 0 and  p (a) = 0 the polynomials  p (x) and q (x) have a common factor (x - a).
The rational function has the removable discontinuity or the hole in the graph at the point  x = a.
When both the numerator and denominator of a rational function vanish at the given point a, we factor and cancel common factors and then find the limit of the equivalent function.
 Example:  Evaluate the limit
 Solution:  To avoid the indeterminate form 0/0, the expression takes as x ® 1, we factor and cancel common factors The rational function has the hole in the graph at x = 1, the vertical asymptote  x = -1 and the horizontal asymptote  y = 2 since as is shown in the right figure.

Calculus contents B