And this right over here mightīe our limit minus epsilon. It's saying look, if we areĬonstraining x in such a way that if x is in thatĪ little bit clearer by diagramming right over here. They want f of x to be to L and the burden is then toįind a delta where as long as x is within deltaĮpsilon of the limit. Which I kind of made as a little bit more of a Get f of x as close as you want to L by making If x is within delta of C, then f of x will be withinĮpsilon of our limit. Positive number which we'll call delta- the lowercaseĭelta, the Greek letter delta- such that where And epsilon is howĬlose do you want to be? How close? So for example, ifĮpsilon is 0.01, that says that you want f of x And you do this by giving meĪ positive number that we call epsilon, which is really So instead of sayingĪs close as you want, let's call that some So let's see if we can putĪ little bit of meat on it.
Of x as close as you want to L by making x Say that the limit of f of x as x approaches C isĮqual L, you're really saying- and this is the somewhat We tried to come up with a somewhat rigorousĭefinition of what a limit is, where we say when you As a footnote, during my school days, another student remarked to me: "Bruce came within epsilon of flunking out last year". Analysis (of which calculus is a part) involves a lot of proving that something is within epsilon of something else.
Remarkably, over such a long time the smartest mathematicians (including Newton) missed discovering that crucial one-liner definition. It is counter-intuitive to math students at first, but after a while they internalize it and it becomes second nature. Many decades after Newton's struggles, the Bohemian mathematician Bolzano turned the question around by asking essentially: "If we want f(x) to get this close to f(c), how close does x have get to c?" This is the basis of the epsilon-delta definition, which finally allowed mathematicians to rigorously prove the calculus invented by Newton (and Leibnitz). The way continuous functions are introduced to beginning calculus students is to ask something intuitive like: "As x gets this close to c, how close does f(x) get to f(c)?” This hand-wavy approach gives students an intuitive feel for the topic, but is not useful for rigorous proofs. Newton could not rigorously prove any of his Calculus despite a lifetime of effort. If ƒ is defined on a set X of real numbers, and if p is a limit point of the intersection of X with (p, +∞), we say that ƒ has right-sided limit L at p if and only if for all ε > 0 there exists δ > 0 such that |ƒ(x) - L| < ε for all x in X with p < x < p + δ. There is a more general notion of one-sided limits. II) One says that ƒ has left-sided limit L at b if and only if there for every real number ε > 0 exists a real number δ > 0 such that |ƒ(x) - L| 0. I will not formulate the most general way of defining one-sided limits (it requires some knowledge of point-set topology), but suppose ƒ is a real-valued function defined on a set containing an open interval of the form (a, b), where a 0 exists a real number δ > 0 such that |ƒ(x) - L| 0.
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