Calculus/Proofs of Some Basic Limit Rules

Formal Definition of the Limit Limits/Exercises 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 . Then when . So when . Formal Definition of the Limit Limits/Exercises