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Decimal Insights on Limits, Continuity, ConvergenceDecimal and decimal-free error-control perspectives of continuity, limits and Cauchy sequences are given below. These perspectives is followed by comments on math education. Continuity and Unlimited Error ControlLimits and continuity in calculus may be described geometrically, that is, intuitively and informally, or more precisely in terms of say epsilons and deltas. The roles of epsilon and delta below are played by E > 0 and D > 0. Imagine for instance we want to compute a function f(x) at the point x = A accurately. So we can ask the error control question how close must x be to A in order for f(x) to agree with f(A) to say k-decimal places. The answer might be that x must agree with A to m decimal places. In some computational problems, this answer for a specified number k of decimal places may be all that is needed. But in other situations, we want in practice or in principle, unlimited error control. Here we may want to say for any k, there is an m such f(x) will agree with f(A) to k decimals if x agrees with A to m decimals. Unlimited error control offers motivation and a perspective on the discussion of continuity. Now will say that f(x) is continuous at x = A if for each whole number k, there is a number m such the limit f(a) and the value of f(x) will agree to k-decimals whenever the number x agrees with the value of a to m decimal places. Continuity here represents the concept of unlimited error control in decimal computations. More generally, we can ask (following Cauchy), given an error control target E > 0, how close must x be to A for the difference of f(x) and f(A) to be less than E in magnitude? The answer follows by obtaining a number D with the property that if |x-A| < d then |f(x) - f(A)| < E. Without reference to decimals we can say that f(x) is continuous at x = A if for every error control tolerance E > 0, there is a number D > 0 such that whenever |x-A| < d then |f(x)- f(A)| < E. Here continuity at x="A" corresponds to the idea of unlimited error control at x="A." This second concept is decimal free. It is traditional to use epsilons and deltas in place of E> 0 and D > 0. LimitsWe will say that a number L is the limit of a function f(x) as x approaches A if one of the following conditions hold:
Both conditions are equivalent. Each implies the other. Why or how depends on how you think of (or represent) the real numbers. For most people, assuming that real numbers are represented by signed decimal expansions (infinite or finite) is sufficient. Modern mathematics has alternate decimal (or base) -free representations of real numbers.
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