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
*b*and*c*are constants then .

**Proof of the Constant Rule for Limits:**

To prove that , 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
*c*is a constant then .

**Proof of the Identity Rule for Limits:**

To prove that , we need to find a such that for every , whenever . Choosing satisfies this condition.

**Scalar Product Rule for Limits**

**Proof of the Scalar Product Rule for Limits:**

Since we are given that , there must be some function, call it , such that for every , whenever . Now we need to find a such that for all , whenever .

First let's suppose that . , so . In this case, letting satisfies the limit condition.

Now suppose that . Since has a limit at , we know from the definition of a limit that is defined in an open interval *D* that contains (except maybe at itself) . In particular, we know that doesn't blow up to infinity within *D* (except maybe at , but that won't affect the limit), so that in *D*. Since is the constant function in *D*, the limit by the Constant Rule for Limits.

Finally, suppose that . , so . In this case, letting satisfies the limit condition.

**Sum Rule for Limits**

Suppose that and . Then

**Proof of the Sum Rule for Limits:**

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 of the Difference Rule for Limits:** Define . By the Scalar Product Rule for Limits, . Then by the Sum Rule for Limits, .

**Product Rule for Limits**

Suppose that and . Then

**Proof of the Product Rule for Limits: ^{[1]}**

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 of the Quotient Rule for Limits:**

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

- 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 of the Squeeze Theorem:**

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 .

## NotesEdit

- ↑ This proof is adapted from one found at planetmath.org/encyclopedia/ProofOfLimitRuleOfProduct.html due to Planet Math user pahio and made available under the terms of the Creative Commons By/Share-Alike License.