Limits at Infinity
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
.