teaching:topics:calculus:limits
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teaching:topics:calculus:limits [2019/06/18 17:12] simon [Limits] |
teaching:topics:calculus:limits [2022/07/26 13:20] (current) simon |
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| - | <WRAP right>~~DIR?noheader~~</WRAP> | ||
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| ====Limits==== | ====Limits==== | ||
| + | \(\newcommand{\defeq}{\stackrel{\scriptsize{\mathsf{def}}}{=}}\) | ||
| To work with these [[sequences#series|infinite sums]] and come to some formal, mathematical, way of resolving the [[sequences#paradox|wolf question]] properly we should have a notation which does not involve '`cdots`' or other vague and perhaps ambiguous ideas. | To work with these [[sequences#series|infinite sums]] and come to some formal, mathematical, way of resolving the [[sequences#paradox|wolf question]] properly we should have a notation which does not involve '`cdots`' or other vague and perhaps ambiguous ideas. | ||
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| We start with a new way of writing the expression for how far the wolf has run after 5 intervals, using a new operation | We start with a new way of writing the expression for how far the wolf has run after 5 intervals, using a new operation | ||
| - | \(\sum\limits_{n=1}^{5}\) called "sum" which means adding together each of the values of the sequence made by setting (in this example) `n` to each integer from 1 to 5 in the expression (or function) after it ... | + | \(\sum\limits_{k=1}^{5}\) called "sum" which means adding together each of the values of the sequence made by setting (in this example) `k` to each integer from 1 to 5 in the expression (or function) after it ... |
| - | \[W_5 = \sum_{n=1}^{5}\frac{200}{2^n}\] | + | \[W_5 = \sum_{k=1}^{5}\frac{200}{2^k}\] |
| - | Then we introduce a new variable, here `i`, like the index or counter in a computer program. Now we can talk about distance travelled after `n` steps ... | + | We are using this new variable, here `k`, like we use the index or counter in procedural computer programs. Now we can talk about distance travelled after counting off `n` intervals in the wolf's pursuit ... |
| - | \[W_n = \sum_{i=1}^n\frac{200}{2^i}\] | + | \[W_n = \sum_{k=1}^n\frac{200}{2^k}\] |
| Lastly we want to be able to write the sum if we kept adding up and somehow (?!?) included **all** of the infinite number of smaller and smaller distances: | Lastly we want to be able to write the sum if we kept adding up and somehow (?!?) included **all** of the infinite number of smaller and smaller distances: | ||
| - | \[W = \sum_{i=1}^\infty\frac{200}{2^i}\] | + | \[W = \sum_{k=1}^\infty\frac{200}{2^k}\] |
| ... then when we think carefully about this ... we certainly want to have`quad W=200 quad`here ... we also want to think about different sums when the answer is not some nice, neat finite value ... and we want to keep our use of \(\large\infty\) carefully contained, where we can make some exact sense of it and we can be more careful about what we mean by phrases like "... will continue endlessly". | ... then when we think carefully about this ... we certainly want to have`quad W=200 quad`here ... we also want to think about different sums when the answer is not some nice, neat finite value ... and we want to keep our use of \(\large\infty\) carefully contained, where we can make some exact sense of it and we can be more careful about what we mean by phrases like "... will continue endlessly". | ||
| So mathematicians invented another notation, and a new ... **very powerful!** ... idea to explore, we use notation like '\(\ \lim\limits_{n\to\infty}\ \)' for the idea of the 'limit' we spoke about above, and start our exploration by defining: | So mathematicians invented another notation, and a new ... **very powerful!** ... idea to explore, we use notation like '\(\ \lim\limits_{n\to\infty}\ \)' for the idea of the 'limit' we spoke about above, and start our exploration by defining: | ||
| - | \[\sum_{i=1}^{\infty}{\rm f}(i) = \lim_{n\to\infty}\sum_{i=1}^n {\rm f}(i)\] | + | \[\large\sum_{k=1}^{\infty}{\rm f}(k) \defeq \lim_{n\to\infty}\sum_{k=1}^n {\rm f}(k)\] |
| ===Now take time to see what we can make of all this!=== | ===Now take time to see what we can make of all this!=== | ||
| - | What do we mean when we write \(\quad\lim\limits_{x\to n}{\rm f}(x)\) ? | + | What do we mean when we write \(\quad\lim\limits_{x\to n}\,{\rm f}(x)\) ? |
| Then ... what does it mean to write \(\quad\lim\limits_{x\to\infty}{\rm f}(x)\) ? | Then ... what does it mean to write \(\quad\lim\limits_{x\to\infty}{\rm f}(x)\) ? | ||
| - | Now ... try to graph the function: \(\displaystyle\quad {\rm f}(x) = \frac{x^2-a^2}{x-a}\) | + | Now ... try to graph the function: \(\displaystyle\quad {\rm f}(x) = \frac{x^2-a^2}{x-a},\quad\) say when `\ a = 2` |
| What is the value of this `text(f)(x)` when`quad x=a` ? | What is the value of this `text(f)(x)` when`quad x=a` ? | ||
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| What about \(\quad\lim\limits_{x\to\infty}{\rm f}(x)\quad\) and \(\quad\lim\limits_{x\to -\infty}{\rm f}(x)\ \)? | What about \(\quad\lim\limits_{x\to\infty}{\rm f}(x)\quad\) and \(\quad\lim\limits_{x\to -\infty}{\rm f}(x)\ \)? | ||
| + | <WRAP right>**=>[[introduction#ideas to consider as you start calculus|motion]]**</WRAP> | ||
teaching/topics/calculus/limits.1560841921.txt.gz · Last modified: 2019/06/18 17:12 by simon
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