OneSided 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
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
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View Full DocumentWe 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 (
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 Fall '08
 sc
 Limits, Limit of a function, Onesided limit, OneSided Limits, Real projective line

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