

The
limit of a function 
Infinite limits 
The
limit of a function examples 
Vertical, horizontal
and slant (or oblique) asymptotes 





Infinite limits 
We
write 

if
f (x) can
be made arbitrarily large by choosing x
sufficiently close
but not equal to a. 
We
write 

if
f (x) can
be made arbitrarily large negative by choosing x
sufficiently close
but not equal to a. 

The
limit of a function examples 
Example: Evaluate
the following limits; 


Solution:
a) As x
tends to minus infinity f
(x) 
gets closer and closer to 0. 
As x
tends to plus infinity f
(x)
gets closer and 
closer to 0. Therefore, 

b)
As x
tends to 0 from the left f
(x)
gets larger in 
negative sense. 



As
x
tends to 0 from the right f(x)
gets larger in positive sense. Therefore, 


Vertical, horizontal
and slant (or oblique) asymptotes 
If
a point (x, y) moves
along a curve f
(x)
and then at least one of its coordinates tends to infinity,
while the distance between the point and a line tends to zero
then, the line is called the asymptote
of the curve. 

Vertical
asymptote 
If
there exists a number a
such that 

then
the line x
= a is the vertical
asymptote. 

Horizontal
asymptote 
If
there exists a number c
such that 

then
the line y
= c is the horizontal
asymptote. 

Slant
or oblique asymptote 
If
there exist limits 

then,
a line y = mx
+ c is the slant
asymptote of the function f
(x). 

Example: Find
the vertical and the horizontal 
asymptote
of the function 

Solution:
Since, 

then x
= 1 is the vertical asymptote. 



Since, 



then y
= 2 is the horizontal asymptote. 








Functions
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