Module 4 : Local / Global Maximum / Minimum and Curve Sketching
Lecture 12 : Asymptoes [Section 12.1]
Thus, is strictly decreasing in and in . Note that, it is not defined at .
Next,
, for all .
Since, for all , and for all , is strictly concave downward in and is strictly concave upward in . Hence, is a point of inflection and the graph of is as follows:
We state next an extension of the idea of a horizontal asymptote.
12.1.4
Definition (Oblique Asymptote) :
(i)
A line is called an oblique asymptote from left to if
,
i.e., the graph approaches the line as approaches .
(ii)
A line is called an oblique asymptote from right to if
,
i.e., the graph approaches the line as approaches .
Note that if , then the oblique asymptote is in fact the horizontal asymptote.
is strictly decreasing in 
and in 
. Note that, it is not defined at 
. 
, for all 
. 
for all 
, and 
for all 
, 
is strictly concave downward in 
and is strictly concave upward in 
. Hence, 
is a point of inflection and the graph of 
is as follows: 
is called an 
if 
, 
approaches the line 
as 
approaches 
. 
is called an 
if 
, 
approaches the line 
as 
approaches 
. 
, then the oblique asymptote is in fact the horizontal asymptote.
