

The
limit of a function 
The
limit of a function examples 
Vertical, horizontal
and slant (or oblique) asymptotes 
Monotone
functions  increasing or decreasing in value 





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. 

Example: Calculate
asymptotes and sketch the graph of the function 


Solution:
By
equating the numerator with zero and solving for
x
we
find the xintercepts, 
x^{2}

x 
2 = (x + 1)(x 
2) = 0, 
x_{1}
= 1
and
x_{2}
= 2. 
We
calculate f
(0)
to find the yintercept, 
f
(0)
= 2/3. 
By
equating the denominator with zero and solving for
x
we find the vertical
asymptote, 
x
= 3. 
Let
calculate following limits 

to
find the slant asymptote y
= mx + c. 





Therefore,
the line y
= x + 2
is the slant asymptote of the given function. 

Example: Find
the following limits, 

Solution:
The graph of the tangent
function shows, 

as
x
approaches
p/2 from the left,
the tangent
function increases
to plus infinity, while as x
approaches
p/2
from the
right, the function decreases to minus infinity, therefore 


Example: Evaluate
the limit 


Solution:




Example: Evaluate
the limits, 


Solution:
The graph of the
arctangent
function shows,
as x
tends to minus infinity the function values
approach 
p/2
while, as x
tends to plus infinity,
the function values approach p/2. 
Therefore, 

and 





Monotone
functions  increasing or decreasing in value 
1.
The function is said to be increasing if f
(x_{1})
< f (x_{2})
for all x_{1}
<
x_{2}. 
2.
The function is said to be decreasing if f
(x_{1})
> f (x_{2})
for all x_{1}
<
x_{2}. 
3.
If f
is either increasing or decreasing then f
is said to be monotone. 









Calculus
contents B 



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