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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# Note a more detailed and rigorous presentation is in

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Unformatted text preview: rst reading it is not necessary to study the Analysis of Limits. The reader may skip the next chapter without loss of continuity. 33 Limits* 9. Analysis of Limits In general, we can talk about the LIMIT of a function f(x) as x → a . Does f(x) TEND TO any par ticular value b as x TENDS TO a ? Can we choose f(x) as close as we like to b b y choosing x sufficiently as s ufficiently close t o a ? Does L imit − f(x) = b = L imit + f(x) ? x→a x→a Does Limit f(a-- δ x) = b = Limit δx → 0 δx → 0 f(a+ δ x) ? If it does, then we call this b the LIMIT of f(x) as x TENDS TO a and we write this as: L imit f(x) = b or L imit f(a+ δ x) = b x→a δx → 0 We are not concerned with if 0 < C f(x)-- bC o r 0 = C f(x) -- b C What we are concerned with is when x → a , for any discrete infinitesimal discrete infinitesimal discr ε a s small as we like : can we have C f(x) -- b C < ε o r equivalently C f(a + δ x) -- b C < ε ? Again, we stress that x = a. f(x) at the instant x = a as we said earlier, is / i nstant Value f(x) = f(a), if it exists. x=a We may combine x → a− and x → a+ into one expression : C x -- aC < δ . Formally, we may define the LIMIT of a function f(x) as x TENDS TO a : Limit f(x) = b if for any discrete infinitesimal ε discrete x→a as small as we like, we can find δ such that : C f(x) -- b C < ε when C x -- aC < δ . *Note : A more detailed and rigorous presentation is in A LITTLE MORE CALCULUS by the author. 34 Based on the formal definition of the “limit of a function” , we may analyse the “limit behaviour of a function near some instant a, and see if it has a Limit or not. Let us look at the example 2 of the previous chapter. Example 2: Exampl e 2: Consider the function f(x) = x + a What is the L imit o f f(x) as x TENDS TO a ? As x → a w e have f(x) → 2 a / Since 0 < C x− a C w e have 0 < C f(x)− 2a C , i.e. f(x) = 2 a. We can choose f(x) as close as we like to 2a by choosing x suf f iciently sufficiently as close we like to s uf uff iciently close necessary c lose t o a. S uf f ic...
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