Timeline for Assessing effectiveness of (epsilon, delta) definitions
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Jul 16, 2017 at 16:41 | comment | added | Deane Yang | It is common for us to justify the $\epsilon$-$\delta$ explanation of limits in terms of error tolerance in practical situations. However, there is a key difference. Engineers don't need the first quantifier. They have an a priori value of $\epsilon$ in mind and solve for the $\delta$. They do not need to solve this for every possible value of $\epsilon$. | |
| Jul 11, 2017 at 9:22 | comment | added | Mikhail Katz | @Ian, that's a useful heuristic. Another useful heuristic is in terms of the "trough/target" dichotomy, which I use when I teach this based on infinitesimals as well. | |
| Jul 10, 2017 at 18:29 | comment | added | Ian | This answer reminds me of how I was taught limits the first time around, which was a kind of middle ground between "total handwaving" and rigorous $\epsilon,\delta$ definitions. Namely I was told, only in words, something along the lines of "$f(x) \to L$ as $x \to a$ if you can make $f(x)$ arbitrarily close to $L$ by taking $x$ arbitrarily close to $a$". This is not 100% unambiguous, but I found it both enlightening and intelligible at the time. | |
| Feb 27, 2014 at 18:20 | comment | added | user44143 | @Steven Gubkin, I agree with your comment about error analysis -- but I would emphasize that you need a uniform $\epsilon-\delta$ analysis for that application. E.g.: What control on the radius do you need to guarantee an error within 1% on the volume? The application is not good with an answer depending on $r$, the actual and unknown radius; it's better to give the answer from uniform continuity that applies to a range of $r$'s. | |
| Feb 27, 2014 at 18:05 | comment | added | Steven Gubkin | As I note in my comment above, it is exactly engineers who would profit the most from serious attention to $\epsilon-\delta$ in the form of rigorous error analysis. In every application, you need to know how good a control on inputs you need to guarantee a given error tolerance on the outputs. Without this bridges collapse, and airplanes fail to get off the ground. | |
| Feb 27, 2014 at 16:34 | history | answered | Lev Borisov | CC BY-SA 3.0 |