In the final two examples in the previous section we saw two limits that did not
exist. However, the reason for each of
the limits not existing was different for each of the examples.
We saw that
did not exist because the function did not settle down to a
single value as t approached . The closer to we moved the more wildly the function
oscillated and in order for a limit to exist the function must settle down to a
single value.
However we saw that
did not exist not because the function didn’t settle down to
a single number as we moved in towards ,
but instead because it settled into two different numbers depending on which
side of we were on.
In this case the function was a very well behaved function,
unlike the first function. The only
problem was that, as we approached ,
the function was moving in towards different numbers on each side. We would like a way to differentiate between
these two examples.
We do this with onesided
limits. As the name implies, with
onesided limits we will only be looking at one side of the point in
question. Here are the definitions for
the two one sided limits.
Righthanded limit
We say
provided we can make f(x)
as close to L as we want for all x sufficiently close to a and x>a without actually letting x be a.

Lefthanded limit
We say
provided we can make f(x)
as close to L as we want for all x sufficiently close to a and x<a without actually letting x be a.

Note that the change in notation is very minor and in fact
might be missed if you aren’t paying attention.
The only difference is the bit that is under the “lim” part of the
limit. For the righthanded limit we now
have (note the “+”) which means that we know
will only look at x>a. Likewise for the lefthanded limit we have (note the “”) which means that we will
only be looking at x<a.
Also, note that as with the “normal” limit (i.e. the limits from the previous
section) we still need the function to settle down to a single number in order
for the limit to exist. The only
difference this time is that the function only needs to settle down to a single
number on either the right side of or the left side of depending on the onesided limit we’re dealing
with.
So when we are looking at limits it’s now important to pay
very close attention to see whether we are doing a normal limit or one of the
onesided limits. Let’s now take a look
at the some of the problems from the last section and look at onesided limits
instead of the normal limit.
Example 1 Estimate
the value of the following limits.
Solution
To remind us what this function looks like here’s the
graph.
So, we can see that if we stay to the right of (i.e.
) then the function is moving in
towards a value of 1 as we get closer and closer to ,
but staying to the right. We can
therefore say that the righthanded limit is,
Likewise, if we stay to the left of (i.e )
the function is moving in towards a value of 0 as we get closer and closer to
,
but staying to the left. Therefore the
lefthanded limit is,
In this example we do get onesided limits even though the
normal limit itself doesn’t exist.

Example 2 Estimate
the value of the following limits.
Solution
From the graph of this function shown below,
we can see that both of the onesided limits suffer the
same problem that the normal limit did in the previous section. The function does not settle down to a
single number on either side of . Therefore, neither the lefthanded nor the
righthanded limit will exist in this case.

So, onesided limits don’t have to exist just as normal
limits aren’t guaranteed to exist.
Let’s take a look at another example from the previous
section.
Example 3 Estimate
the value of the following limits.
Solution
So as we’ve done with the previous two examples, let’s
remind ourselves of the graph of this function.
In this case regardless of which side of we are on the function is always approaching
a value of 4 and so we get,

Note that onesided limits do not care about what’s
happening at the point any more than normal limits do. They are still only concerned with what is
going on around the point. The only real
difference between onesided limits and normal limits is the range of x’s that we look at when determining the
value of the limit.
Now let’s take a look at the first and last example in this
section to get a very nice fact about the relationship between onesided limits
and normal limits. In the last example
the onesided limits as well as the normal limit existed and all three had a
value of 4. In the first example the two
onesided limits both existed, but did not have the same value and the normal
limit did not exist.
The relationship between onesided limits and normal limits
can be summarized by the following fact.
Fact
Given a function f(x)
if,
then the normal limit will exist and
Likewise, if
then,

This fact can be turned around to also say that if the two
onesided limits have different values, i.e.,
then the normal limit will not exist.
This should make some sense.
If the normal limit did exist then by the fact the two onesided limits
would have to exist and have the same value by the above fact. So, if the two onesided limits have
different values (or don’t even exist) then the normal limit simply can’t
exist.
Let’s take a look at one more example to make sure that
we’ve got all the ideas about limits down that we’ve looked at in the last
couple of sections.
Hopefully over the last couple of sections you’ve gotten an
idea on how limits work and what they can tell us about functions. Some of these ideas will be important in
later sections so it’s important that you have a good grasp on them.