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 Properties section of the Extras chapter.
First we will assume that and exist and that c is any constant. Then,
In other words we can “factor” a
multiplicative constant out of a limit.
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 “-”.
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.
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.
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.
can be done for any integer n.
This is just a special case of
the previous example.
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 .
As with the last one you should
be able to convince yourself of this by drawing the graph of .
This is really
just a special case of property 5
Note that all these properties also hold for the two
one-sided limits as well we just didn’t write them down with one sided limits
to save on space.
Let’s compute a limit or two using these properties. The next couple of examples will lead us to
some truly useful facts about limits that we will use on a continual basis.
Example 1 Compute
the value of the following limit.
This first time through we will use only the properties
above to compute the limit.
First we will use property 2 to break up the limit into three separate limits. We will then use property 1 to bring the constants out of the
first two limits. Doing this gives us,
We can now use properties 7 through 9 to
actually compute the limit.
Now, let’s notice that if we had defined
then the proceeding example would have been,
In other words, in this case we see that the limit is the same
value that we’d get by just evaluating the function at the point in
question. This seems to violate one of
the main concepts about limits that we’ve seen to this point.
In the previous two sections we made a big deal about the
fact that limits do not care about what is happening at the point in
question. They only care about what is
happening around the point. So how does
the previous example fit into this since it appears to violate this main idea
Despite appearances the limit still doesn’t care about what
the function is doing at . In this case the function that we’ve got is
simply “nice enough” so that what is happening around the point is exactly the
same as what is happening at the point. Eventually we will formalize up just what is meant by
“nice enough”. At this point let’s not
worry too much about what “nice enough” is.
Let’s just take advantage of the fact that some functions will be “nice
enough”, whatever that means.
The function in the last example was a polynomial. It turns out that all polynomials are “nice
enough” so that what is happening around the point is exactly the same as what
is happening at the point. This leads to
the following fact.
If p(x) is a
By the end of this section we will generalize this out
considerably to most of the functions that we’ll be seeing throughout this
Let’s take a look at another example.
Example 2 Evaluate
the following limit.
First notice that we can use property 4) to write the limit as,
Well, actually we should be a little careful. We can do that provided the limit of the
denominator isn’t zero. As we will see
however, it isn’t in this case so we’re okay.
Now, both the numerator and denominator are polynomials so
we can use the fact above to compute the limits of the numerator and the
denominator and hence the limit itself.
Notice that the limit of the denominator wasn’t zero and
so our use of property 4 was
Notice in this last example that again all we really did was
evaluate the function at the point in question.
So it appears that there is a fairly large class of functions for which
this can be done. Let’s generalize the
fact from above a little.
Provided f(x) is
“nice enough” we have,
Again, we will formalize up just what we mean by “nice
enough” eventually. At this point all we want to do is worry
about which functions are “nice enough”.
Some functions are “nice enough” for all x while others will only be “nice enough” for certain values of x.
It will all depend on the function.
As noted in the statement, this fact also holds for the two
one-sided limits as well as the normal limit.
Here is a list of some of the more common functions that are
are nice enough for all x’s.
- If then f(x)
will be nice enough provided both p(x)
and q(x) are nice enough and if
we don’t get division by zero at the point we’re evaluating at.
- are nice enough for all x’s
- are nice enough provided In other words secant and tangent are
nice enough everywhere cosine isn’t zero.
To see why recall that these are both really rational functions and
that cosine is in the denominator of both then go back up and look at the
second bullet above.
- are nice enough provided In other words cosecant and cotangent
are nice enough everywhere sine isn’t zero.
- is nice enough for all x if n is odd.
- is nice enough for if n
is even. Here we require to avoid having to deal with complex
- are nice enough for all x’s.
- are nice enough for x>0. Remember we can
only plug positive numbers into logarithms and not zero or negative
sum, difference or product of the above functions will also be nice
enough. Quotients will be nice
enough provided we don’t get division by zero upon evaluating the limit.
The last bullet is important. This means that for any combination of these
functions all we need to do is evaluate the function at the point in question,
making sure that none of the restrictions are violated. This means that we can now do a large number
Example 3 Evaluate
the following limit.
This is a combination of several of the functions listed
above and none of the restrictions are violated so all we need to do is plug
in into the function to get the limit.
Not a very pretty answer, but we can now do the limit.