One of our fundamental concepts is the limit of a function at a point. But soon we’ll need to consider what happens as we let the input to a function grow without bound.
So let’s consider a function 
defined for 
, where this interval means the set 
. It really doesn’t matter here what 
is, just that we’ve got some point where 
is defined for all larger numbers. We want to come up with a sensible definition for 
.
When we took a limit at a point 
we said that 
if for every 
there is a 
so that 
implies 
. But this talk of 
and 
is all designed to stand in for neighborhoods in a metric space. Picking a defines a neighborhood of the point . All we need is to come up with a notion of a “neighborhood” of 
.
What we’ll use is a ray just like the one above: 
. This seems to make sense as the collection of real numbers “near” infinity. So let’s drop it into our definition: the limit of a function at infinity, is 
if for every there is an 
so that 
implies . It’s straightforward to verify from here that this definition of limit satisfies the same laws of limits as the earlier definition.
Finally, we can define neighborhoods of 
as leftward rays 
. Then we get a similar definition of the limit of a function at .
One particular limit that’s useful to have as a starting point is 
. Indeed, given we can set 
. Then if 
we see that 
, establishing the limit.
From here we can handle the limit at infinity of any rational function 
. Let’s split off the top degree terms from the polynomials 
and 
. Divide through top and bottom by 
to write
Now every term in 
has degree less than 
, so each is a multiple of some power of 
. The laws of limits then tell us that they go to 
, and the limit of the denominator of is 
. Thus our limit is the limit of the numerator.
If 
we have a positive power of 
as our leading term, which goes up to or down to (depending on the sign of 
. If 
, all the powers are negative, and thus the limit is . And if 
, then all the other powers are negative, and the limit is .
So if the numerator of has the higher degree, we have 
. If the denominator has higher degree, then 
. If the degrees are equal, we compare the leading coefficients and find 
.
April 17, 2008
Posted by John Armstrong |
Analysis, Calculus |
4 Comments
