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 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

Suppose that for finite and that is constant. Then

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

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 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 .

Notes[edit]

  1. 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.
← Formal Definition of the Limit Calculus Limits/Exercises →
Proofs of Some Basic Limit Rules