Infinite Limits
In this section we will take a look at limits whose value
is infinity or minus infinity. These kinds of limit will show up fairly
regularly in later sections and in other courses and so you�ll need to be able
to deal with them when you run across them.
The first thing we should probably do here is to define
just what we mean when we sat that a limit has a value of infinity or minus
infinity.
Definition
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We say
if we can make f(x) arbitrarily large for
all x sufficiently close to x=a, from both sides, without
actually letting
 .
We say
if we can make f(x) arbitrarily large and
negative for all x sufficiently close to x=a, from both
sides, without actually letting
 .
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These definitions can be appropriately modified for the
one-sided limits as well. To see a more precise and mathematical definition of
this kind of limit see the
The Definition of the Limit section at the end of this chapter.
Let�s start off with a fairly typical example illustrating
infinite limits.
Example 1
Evaluate each of the following limits.
Solution
So we�re going to be taking a look at a couple of one-sided
limits as well as the normal limit here. In all three cases notice that we
can�t just plug in
 .
If we did we would get division by zero. Also recall that the definitions above
can be easily modified to give similar definitions for the two one-sided limits
which we�ll be needing here.
Now, there are several ways we could proceed here to get
values for these limits. One way is to plug in some points and see what value
the function is approaching. In the proceeding section we said that we were no
longer going to do this, but in this case it is a good way to illustrate just
what�s going on with this function.
So, here is a table of values of x�s from both the
left and the right. Using these values we�ll be able to estimate the value of
the two one-sided limits and once we have that done we can use the
fact that the normal limit will exist only if the two one-sided limits exist
and have the same value.
|
x |
|
x |
|
|
-0.1 |
-10 |
0.1 |
10 |
|
-0.01 |
-100 |
0.01 |
100 |
|
-0.001 |
-1000 |
0.001 |
1000 |
|
-0.0001 |
-10000 |
0.0001 |
1000 |
From this table we can see that as we make x smaller
and smaller the function
 gets
larger and larger and will retain the same sign that x originally had.
It should make sense that this trend will continue for any smaller value of x
that we chose to use. The function is a constant (one in this case) divided by
an increasingly small number. The resulting fraction should be an increasingly
large number and as noted above the fraction will retain the same sign as x.
We can make the function as large and positive as we want
for all x�s sufficiently close to zero while staying positive (i.e.
on the right). Likewise, we can make the function as large and negative as we
want for all x�s sufficiently close to zero while staying negative (i.e.
on the left). So, from our definition above it looks like we should have the
following values for the two one sided limits.
Another way to see the values of the two one sided limits
here is to graph the function. Again, in the previous section we mentioned that
we won�t do this too often as most functions are not something we can just
quickly sketch out as well as the problems with accuracy in reading values off
the graph. In this case however, it�s not too hard to sketch a graph of the
function and, in this case as we�ll see accuracy is not really going to be an
issue. So, here is a quick sketch of the graph.
So, we can see from this graph that the function does
behave much as we predicted that it would from our table values. The closer
x gets to zero from the right the larger (in the positive sense) the
function gets, while the closer x gets to zero from the left the larger
(in the negative sense) the function gets.
Finally, the normal limit, in this case, will not exist
since the two one-sided have different values.
So, in summary here are the values of the three limits for
this example.
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