One-Sided Limits
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 one-sided limits. As the name
implies, with one-sided limits we will only be looking at one side of the point
in question. Here are the definitions for the two one sided limits.
Right-handed 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. |
Left-handed 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 right-handed limit we now
have
 (note
the �+�) which means that we know will only look at x>a. Likewise for
the left-handed 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 one-sided 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
one-sided limits. Let�s now take a look at the some of the problems from the
last section and look at one-sided limits instead of the normal limit.
Example
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 right-handed 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 left-handed limit is,
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