
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. 


Evaluate
the limit 



Solution:





Evaluate
the limit 



Solution:





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



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



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. 













Solved
problems contents  A 



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