Two of the functions we looked at in Lesson 2, and , also have seemingly interesting behavior as x increases without bound. Upon looking at the graphs of and , it seems that y is getting very close to zero as x is getting very large. We can write this mathematically as . It is also apparent that .
The graph of
The graph of
Let’s also consider and . Upon looking at the graphs of , we might conclude that . The graph of is a bit more interesting. Firstly, we note that the function’s ;limit is undefined at . (Note that in calculus, we always measure angles in radians.) However, the limit from the right is and the limit from the left is . Thus we must write and . In fact, since is periodic with period , we can say and for (i.e., k is an integer).
and 
, also have seemingly interesting behavior as x increases without bound. Upon looking at the graphs of 
and 
, it seems that y is getting very close to zero as x is getting very large. We can write this mathematically as 
. It is also apparent that 
.
and 
. Upon looking at the graphs of 
, we might conclude that 
. The graph of 
is a bit more interesting. Firstly, we note that the function’s ;limit is undefined at 
. (Note that in calculus, we always measure angles in radians.) However, the limit from the right is 
and the limit from the left is 
. Thus we must write 
and 
. In fact, since 
is periodic with period 
, we can say 
and 
for 
(i.e., k is an integer).
