Limit Properties
The time has almost come for us to actually compute some
limits. However, before we do that we will need some properties of limits that
will make our life somewhat easier. So, let�s take a look at those first. The
proof of some of these properties can be found in the
Proof of Various Limit Propertiessection of the Extras chapter.
Properties
First we will assume that
 and
 exist
and that c is any constant. Then,
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In other words we can �factor�
a multiplicative constant out of a limit.
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So to take the limit of a sum
or difference all we need to do is take the limit of the individual parts and
then put them back together with the appropriate sign. This is also not limited
to two functions. This fact will work no matter how many functions we�ve got
separated by �+� or �-�.
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We take the limits of products
in the same way that we can take the limit of sums or differences. Just take
the limit of the pieces and then put them back together. Also, as with sums or
differences, this fact is not limited to just two functions.
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As noted in the statement we
only need to worry about the limit in the denominator being zero when we do the
limit of a quotient. If it were zero we would end up with a division by zero
error and we need to avoid that.
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In this property n can
be any real number (positive, negative, integer, fraction, irrational, zero,
etc.). In the case that n is an integer this rule can be thought of
as an extended case of 3.
For example consider the case
of n = 2.
The same can be done for any integer n.
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This is just a special case of
the previous example.
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In other words, the limit of a
constant is just the constant. You should be able to convince yourself of this
by drawing the graph of
 .
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As with the last one you should
be able to convince yourself of this by drawing the graph of
 .
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This is
really just a special case of property 5 using
 .
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