ALGEBRA - solved problems
The limit of a function
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.
349.
   Evaluate the limit
Solution:  
350.
   Evaluate the limit
Solution:  
351.
   Evaluate the limit
Solution:  
Evaluating the limit of a rational function at a point
a)  The limit of a rational function that is defined at the given point
352.
   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
353.
   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.
 
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