Module 4 : Local / Global Maximum / Minimum and Curve Sketching
Lecture 12 : Asymptoes [Section 12.1]
12.1
Asymtotes
In this section we analyze the behavior of a function when and the situations when . We first look at an example.
12.1.1
Examples:
Consider the function
Since, for every , the graph of the function lies above the -axis.
Further, and , for all .
Thus, is a strictly decreasing, concave up function. Thus, it is natural to ask, what does the graph look like for all large values of , and for all small values of ? We note that and .
We can interpret these as follows: as increases to , comes closer and closer to -axis, the line , and as decreases to 0, comes closer and closer to -axis, the line . Thus the graph of is as follows:
when 
and the situations when 
. We first look at an example. 
for every 
, the graph of the function lies above the 
-axis. 
and 
, for all 
. 
is a strictly decreasing, concave up function. Thus, it is natural to ask, what does the graph look like for all large values of 
, and for all small values of 
? We note that 
and 
. 
increases to 
, 
comes closer and closer to 
-axis, the line 
, and as 
decreases to 0, 
comes closer and closer to 
-axis, the line 
. Thus the graph of 
is as follows: 
