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Unformatted text preview: Lecture 10 Mathematical definition of limit (cont’d) We shall now use the formal “ ǫ,δ ”definition of the limit, given in the previous lecture, to prove a couple of mathematical theorems involving limits, namely (i) the sum rule for limits and (ii) the “Squeeze Theorem”. The purpose of this exercise is to give you an idea of how mathematics is really done, i.e., by starting with some precise mathematical definitions and then, by means of mathematical logic, arriving at some new results, in the form of “theorems.” It is also important to note that a “theorem” is usually formulated in terms of one or more assumptions or hypotheses , from which a result will be derived. The generic form in which a theorem is stated is as follows, If (hypotheses) then (conclusion). In the notes below, we shall use the symbols “ H1 ”, “ H2 ”, etc., to denote the hypotheses in a theorem and “ C ” to denote the conclusion. (This was not done in class.) A disclaimer: Please also note that the proofs that are presented below are, as was done in the lectures, in a quite “expanded” form, containing explanatory paragraphs and side comments, in order to give you an idea of how and why certain things are done – and often the actual order of thinking that goes on in the development of a proof. From a mathematician’s point of view, they are quite verbose and can certainly be “trimmed down.” In fact, you will be referred to the “leaner and meaner” mathematical versions of the theorems as they appear in the text. Sum law for limits We now proceed to prove the following “limit law” in terms of the formal definition of a limit. We shall write it in the “hypothesis/conclusion” form mentioned above: Theorem: If H1: lim x → a f ( x ) = L and H2: lim x → a g ( x ) = M, (1) 61 then C: lim x → a [ f ( x ) + g ( x )] = L + M. (2) Comments: Before going on with the proof, let’s step back for a moment. Since we’re going to use the ǫ,δ definition of limit, we’ll probably be using the fact that “For any ǫ , there exists a δ ” for each of the limits in H1 and H2 . And we’ll eventually have to show that such an ǫ,δ condition holds for the limit in C . If we let h ( x ) = f ( x ) + g ( x ) and H = L + M, (3) then we’re going to have to show that, for any ǫ > 0, there exists a δ > 0 such that  h ( x ) − H  < ǫ for all x such that 0 <  x − a  < δ. (4) In terms of f and g , this becomes,  f ( x ) + g ( x ) − ( L + M )  < ǫ for all x such that 0 <  x − a  < δ. (5) The goal of our proof is to show that the LHS of this inequality can be made less than some ǫ , which then means that we can “push” it toward zero....
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 Fall '08
 SPEZIALE
 Calculus, Limits, Limit, lim, Continuous function, Inverse function

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