The goal is to choose a point (x,y) on the graph such that y is within ε of L. The means is to specify δ so that any x within δ of x0 (except possibly x = x0) will cause y = f(x) to work: to be within ε of L.
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Both the derivative and the definite integral are defined by certain limits. Working informally, the idea of limit is a relatively small step up the mathematical ladder. Formally, the definition of limit is more technical. It is not so hard to see why mathematicians struggled with this concept. The first to write down the equivalent of the modern definition of limit was Heine, influenced by Weierstrass, Cauchy, and others.
Let (a,b) be an open interval containing the number x0. Let f be a function defined in (a,b), except possibly at x0. Here is what it means for f to have a limit at x0.
The idea:
Roughly, the limit is the value that f(x) should assume when x = x0. It is a prediction based on nearby x-values.
Less roughly, to say the limit is L, you have this situation:
- The goal is to make f(x) close to L, as close as you might want. But f(x) = L is not necessary.
- The means is to let x get close to x0, as close as you need to. But x = x0 is not allowed.
In words:
The limit of f as x approaches x0 is L if for every y-distance ε there is an x-distance δ such that if x is δ-close to x0 (but not equal to x0), then f(x) is ε-close to L.
In a picture:
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The goal is to choose a point (x,y) on the graph such that y is within ε of L. The means is to specify δ so that any x within δ of x0 (except possibly x = x0) will cause y = f(x) to work: to be within ε of L. |
In symbols:
f(x) = L if for all ε > 0 there exists δ > 0 such that 0 < |x-x0| < δ implies | f(x) - L| < ε.
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