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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# Will x b no 0 cxbc what is the difference between x

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Unformatted text preview: at x TENDS TO 0.5, we write this as: x → a . Will x = a ? No. We will always have 0 < C x −a C L I M I T: If we let b = 0.51 we are correct in saying: x TENDS TO 0.51, denoted by x → b. Will x = b ? No. 0 < Cx−bC What is the difference between x → a and x → b ? In x → b : we cannot choose x as close as we like to b . The difference as close we like to we like as C x− bC c annot be made as small as w e lik e . We cannot have C x −bC < δ for any small quantity δ as small as we like . For example: as if we choose δ = 0 .001 we will have 0 < δ < 0 .01 < C x − bC . 14 In x → a : we can choose x as close as we like to a. The difference C x− aC c an as to be made as small as we like . If we choose δ > 0 , any small positive as quantity as small as we like , we can choose x such that C x − aC < δ . as Since we can choose x as close as we like to a , as to and only to a but not to any other point , we say: p oint LIMIT as x → a i s a , written as L imit x = a x→a In x = 0.4, 0.49, 0.499, 0.4999, ... progressively with a = 0.5, since x < a, we can be more precise and say x → a from the left. We write this as x → a--. We could have let x = 0.51, 0.501, 0.5001, 0.50001, ... progressively with a = 0.5 . Here too we have x → a but from the right We write this as x → a + . right. Sometimes we will be very specific and say: x TENDS TO a from the left written as : x → a -left, x TENDS TO a from the right written as : x → a+ right, Which ever be the case, whether x → a+ ( from the right or x → a -- (from right) the left we insist that x ≠ a, i.e. 0 < C x−aC . The case when x = a we will left) speak of separately. L Since we know that LIMIT (x → a − ) is a , we write : x imita-- { x } = a . → Since we know that LIMIT (x → a + ) is a , we write : L imita+ { x } = a . x→ x = 0.4, 0.49, 0.499, ... ...,0.5001, 0.501, 0.51= x a +→ x x → a -a=0.5 − 3 − 2 − 1 0 15 1 2 3 5 . INFINITESIMALS Let us now focus on the dif...
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