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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

Ell defined also the functions we deal with must take

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Unformatted text preview: x = a we have two corresponding CHANGES δ y. f(x) which δ y do we choose ? δx x a a + δx (0, 0) We may make f(x) SINGLE VALUED by restricting the range as in either one of the diagrams below. f(x) f(x) δy δx δy δx (0, 0) a a + δx x (0, 0) 22 a a + δx x The function y = √x for x ≥ 0 is not SINGLE VALUED. It has two values: +√ x and − √x . We must specify which root we are using. y y = +√ x δx (0, 0) δy > 0 a+δx a δx x δy < 0 y = −√ x Since we are dealing with real valued functions over the real line we avoid cases real real where the function may take on complex values such as √x for x < 0. ell-defined Also, the functions we deal with must take on well-defined real values. We avoid undefined entities such as +∞, −∞, division by zero and ∞/∞. Functions of the form a n x n + a n-- 1 x n--1 + . . . + a 2 x 2 + a 1 x + a 0 a re called polynomials . These functions are single valued. p olynomials single 23 8. L I M I T o f a F u n c t i o n To understand the behaviour of a function f(x) very close to some instant a , that instant is to say near instant a and to the LEFT and RIGHT of instant a , we must instant instant analyse : L imit { f(a − δ x) } L imit { f(x) } or δx → 0 x → a-- L imit { f(x) } and x → a+ or L imit { f(a + δ x) } δx → 0 If it turns out that there is some definitive real number b such that : LIMIT from the LEFT = b = LIMIT from the RIGHT L imit { f(x) } =b= L imit { f(a − δ x) } =b= x → a-δx → 0 then we say : L imit { f(x) } x → a+ L imit { f(a + δ x) } δx → 0 L imit { f(x) } = b . x→a Properly speaking, finding or calculating the xLimit+ { f(x) } is a 2 step process. →a This becomes clear when we use the infinitesimal δ x. step 1. TENDS TO step : as x → a+ we may say x = a + δ x. We may substitute (a + δ x) for x in f(x). Then we do whatever expansion, regrouping of terms, cancellation and simplification possible. We may perform these operations because δ x ≠ 0 . It only TENDS TO zero. There is no division by...
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