teaching:topics:calculus:limits
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teaching:topics:calculus:limits [2019/10/03 15:40] simon |
teaching:topics:calculus:limits [2022/07/26 13:20] (current) simon |
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| - | <WRAP right>~~DIR?noheader~~</WRAP> | ||
| - | |||
| ====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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| \(\sum\limits_{k=1}^{5}\) 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 ... | \(\sum\limits_{k=1}^{5}\) 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 ... | ||
| \[W_5 = \sum_{k=1}^{5}\frac{200}{2^k}\] | \[W_5 = \sum_{k=1}^{5}\frac{200}{2^k}\] | ||
| - | We are using this new variable, here `k`, like we use the index or counter in a computer program. Now we can talk about distance travelled after counting off `n` intervals in the wolf's pursuit ... | + | 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 ... |
| \[W_n = \sum_{k=1}^n\frac{200}{2^k}\] | \[W_n = \sum_{k=1}^n\frac{200}{2^k}\] | ||
| 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: | ||
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| 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: | ||
| - | \[\sum_{k=1}^{\infty}{\rm f}(k) = \lim_{n\to\infty}\sum_{k=1}^n {\rm f}(k)\] | + | \[\large\sum_{k=1}^{\infty}{\rm f}(k) \defeq \lim_{n\to\infty}\sum_{k=1}^n {\rm f}(k)\] |
| ===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> | ||
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