Calculus/Proofs of Some Basic Limit Rules

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← Formal Definition of the Limit Calculus Limits/Exercises →
Proofs of Some Basic Limit Rules

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.


Scalar Product Rule for Limits
Suppose that for finite and that is constant. Then
Proof

Given the limit above, there exists in particular a such that whenever , for some such that . Hence


Sum Rule for Limits

Suppose that and . Then

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 .


Difference Rule for Limits

Suppose that and . Then

Proof

Define . By the Scalar Product Rule for Limits, . Then by the Sum Rule for Limits, .


Product Rule for Limits

Suppose that and . Then

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


Quotient Rule for Limits

Suppose that and and . Then

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


Theorem: (Squeeze Theorem)
Suppose that holds for all in some open interval containing , except possibly at itself. Suppose also that . Then also.
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 .
or
when .
So
when .

Notes[edit]


← Formal Definition of the Limit Calculus Limits/Exercises →
Proofs of Some Basic Limit Rules