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May 18th, 2018, 10:30 PM   #1
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Newton's method vs. limit definition

Can Newton's method be effective in defining limits with an epsilon-delta form?

Or, is it relatively intractable for intricate, yet linear curves upon approaching the infinitesimal level -- more than the traditional definition?
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May 19th, 2018, 02:53 AM   #2
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Frankly I don't understand what you are asking. What I know as "Newton's method" is a way of numerically solving an equation such as f(x)= 0 by choosing a point, $\displaystyle x_0$, approximating f by a linear function, determining $\displaystyle x_1$ where that line crosses the x-axis, then iterating. It has nothing to do with "epsilon-delta" and limits. And I simply cannot understand how a "linear curve" could be "intricate"! There is nothing less "intricate" than a line. And what "traditional method' (of what?) are you talking about?
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May 19th, 2018, 11:57 AM   #3
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"Augustin-Louis Cauchy in 1821...followed by Karl Weierstrass, formalized the definition of the limit of a function which became known as the (ε, δ)-definition of limit" [Wiki].

Say a limit point is focused upon by smaller and smaller intervals of domain, and in respect, range. Similarly, Newton's method approaches the value of a function by successively closer approximations, using tangents to the curve.

However, the function whose value we are trying to find by Newton's method may be separated from the initial approximation by values that do not allow his method to be used (maybe by a function that varies faster than the tangent along its curve) so when we choose a f(x) close to the sought-after value, it is in turn no guarantee that its tangent would bring us any closer.

I suppose I am mistaking the iteration of Newton's method calculating a function for the general approach to a constant in the definition of a limit.
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May 19th, 2018, 12:42 PM   #4
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Quote:
Originally Posted by Loren View Post
"Augustin-Louis Cauchy in 1821...followed by Karl Weierstrass, formalized the definition of the limit of a function which became known as the (ε, δ)-definition of limit" [Wiki].

Say a limit point is focused upon by smaller and smaller intervals of domain, and in respect, range. Similarly, Newton's method approaches the value of a function by successively closer approximations, using tangents to the curve.

However, the function whose value we are trying to find by Newton's method may be separated from the initial approximation by values that do not allow his method to be used (maybe by a function that varies faster than the tangent along its curve) so when we choose a f(x) close to the sought-after value, it is in turn no guarantee that its tangent would bring us any closer.

I suppose I am mistaking the iteration of Newton's method calculating a function for the general approach to a constant in the definition of a limit.
Well I don't think Wiki quite says that.

https://en.wikipedia.org/wiki/(%CE%B...ition_of_limit

Although it is true that you must already know quite a lot about the subject to understand what Wiki does say.

I agree with CountryBoy that it is not at all clear what you are trying to ask.

Note that the modern definition of a limit point is in terms of neighborhoods and convergent sequences.
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May 19th, 2018, 01:01 PM   #5
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So you mean "Newton's method" of defining the derivative. That was not clear. Newton had "methods" for doing a variety of things!
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May 26th, 2018, 09:14 AM   #6
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Isaac Newton developed the "derivative" (fluxions) in terms of "infinitesmals" but never said precisely what that meant. The Bishop Berkeley famously satirized them as "ghosts of departed quantities". However, in the 1960s, Abraham Robinson developed "non-standard Analysis" (https://en.wikipedia.org/wiki/Non-standard_analysis), extending the real number system so that it included "infinitesmals" as well as infinite numbers.
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