Limits of rational functions
         Evaluating the limit of a rational function at infinity
         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
Limits of rational functions
A rational function is the ratio of two polynomial functions
where n and m define the degree of the numerator and the denominator respectively.
Evaluating the limit of a rational function at infinity
To evaluate the limit of a rational function at infinity we divide both the numerator and the denominator of the function by the highest power of x of the denominator.
Example:  Evaluate the limit  
Solution:  
Example:  Evaluate the limit  
Solution:  
Example:  Evaluate the limit  
Solution:  
Therefore, the following three cases are possible;
1. If the degree of the numerator is greater than the degree of the denominator (n > m), then the limit of the rational function does not exist, i.e., the function diverges as x approaches infinity.
2. If the degree of the numerator is equal to the degree of the denominator (n = m), then the limit of the rational function is the ratio an/bm of the leading coefficients.
3. If the degree of the numerator is less than the degree of the denominator (n < m), then the limit of the rational function, as x tends to infinity, is zero.
The line y = c is a horizontal asymptote of the graph of f(x) if  
Thus, a rational function has a horizontal asymptote if the function values tend to a constant value c as x
approaches plus or minus infinity.
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.
Functions contents E
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