

Limit of a function
properties (theorems or laws) 
Squeeze
rule

Composition
rule

Limits of
functions
properties use, examples 
Limits of
rational functions 
Evaluating
the limit of a rational function at infinity 






Limit
properties 
Assume that 

exist
and that c
is any constant. Then, 




11.
Squeeze rule 
If f(x)
< g(x) < h(x)
for all x
in an open interval that contains a,
except possibly at x
= a, 
and 



then 




12.
Composition rule 
If f(x)
is continuous at x
= b
and 



then, 




Limits of
functions
properties use, examples 
Limits
that are commonly used are written below for easy reference, 


By
comparing the area of the sector of the arc x
with areas
of the two right triangles, the smaller triangle
with
legs sin x
and cos x,
and the bigger triangle with legs
tan x
and 1, shown on the right figure, we get 

Since
the left and the right side of the last
inequality tend
to the same limit 1, as x
tends to 0 then, applying
the squeeze rule obtained is 




the
fundamental or basic trigonometric limit.
Note
that the arc length x
is measured in radians. 
Further, 

since both
factors on the right tend to 1, as x
tends to 0, then 


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 a_{n}/b_{m}
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.










Calculus
contents B 



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