Now that we have the formal definition of a limit, we can set about proving some of the properties we stated earlier in this chapter about limits.
Constant Rule for Limits
If
are constants then
.
- Proof of the Constant Rule for Limits
We need to find a
such that for every
,
whenever
.
and
, so
is satisfied independent of any value of
; that is, we can choose any
we like and the
condition holds.
Identity Rule for Limits
If
is a constant then
.
- Proof
To prove that
, we need to find a
such that for every
,
whenever
. Choosing
satisfies this condition.
- Proof
Given the limit above, there exists in particular a
such that
whenever
, for some
such that
. Hence

- Proof
Since we are given that
and
, there must be functions, call them
and
, such that for all
,
whenever
, and
whenever
.
Adding the two inequalities gives
. By the triangle inequality we have
, so we have
whenever
and
. Let
be the smaller of
and
. Then this
satisfies the definition of a limit for
having limit
.
- Proof
Define
. By the Scalar Product Rule for Limits,
. Then by the Sum Rule for Limits,
.
- Proof
Let
be any positive number. The assumptions imply the existence of the positive numbers
such that
when 
when 
when 
According to the condition (3) we see that
when 
Supposing then that
and using (1) and (2) we obtain

- Proof
If we can show that
, then we can define a function,
as
and appeal to the Product Rule for Limits to prove the theorem. So we just need to prove that
.
Let
be any positive number. The assumptions imply the existence of the positive numbers
such that
when 
when 
According to the condition (2) we see that
so
when 
which implies that
when 
Supposing then that
and using (1) and (3) we obtain

- Proof
From the assumptions, we know that there exists a
such that
and
when
.
These inequalities are equivalent to
and
when
.
Using what we know about the relative ordering of
, and
, we have
when
.
Then
when
.
So
when
.