alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

So we can see that as x tends irr ir to a the

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Unformatted text preview: ientl y c lose means as c lose as necessar y . a s close If we want 0 < C f(x)− 2 aC < ε some discrete infintesimal as small as we like, we can choose x sufficiently close to a. For example, we may choose : ε 0 < C x− a C < δ , w here δ = − 2 Work it out: ε ε ε 0 < C x− a C < δ ⇒ 0 < C x− a C < − ⇒ a −− < x < a + − 2 2 2 ε ) + a = 2a + − ε f(x) = x + a = ( a + − -- 2 -- 2 ε ε Now: 0 < C f(x) --2a C = C (2a + − ) -- 2a C = − which is < ε -- 2 2 ε So when C x -- a C < − we will have C f(x) -- 2a C < ε 2 We can say: Limit f(x) = 2a x→a Instead of saying: Limit f(x) which is Limit (x + a) x→a x→a we can use the infinitesimal δ x and say: Limit (x + a) = L imit ((a-- δ x ) + a ) = 2a x → a− δx → 0 and Limit (x+a) = L imit ((a+ δ x)+a) = 2a x → a+ δx → 0 35 (from the left ) left lef (from the right) right Let us now look at the example 9 of the previous chapter. Example 9 : +1 if x is rational rational f(x) = { 0 if x is ir r ational irr ir Limit f(x) = b . Let us try to choose an ε such that C f(x) --bC > ε Suppose the x → a for some x very close to a, that is to say for 0 < C x − a C < δ . Recall what we said about the rationals and irrationals. Between any two rationals there are infinitely many rationals. And between any two rationals there are also rationals irr infinitely many ir r ationals . ir No matter how close x is to a there will always be infinitely many rational rational numbers and infinitely many ir r ational numbers. So we can see that as x TENDS irr ir TO a, the function f(x) will keep fluctuating : f(x) will be 0 or 1. There are two possibilities : b = 0 or b ≠ 0. Case b = 0 : choose ε = 1/2 No matter how close x is to a , that is to say no matter how small the δ w e choose, there will always be some r a tional n umbers such that 0 < C x− aC < δ . If x is rational then C f(x) --bC = C 1 --0C = 1 > ε = 1/2 . So b = 0 cannot...
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