This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision | ||
teaching:topics:calculus:limits [2020/09/13 18:53] simon |
teaching:topics:calculus:limits [2022/07/26 13:20] (current) simon |
||
---|---|---|---|
Line 8: | Line 8: | ||
\(\sum\limits_{k=1}^{5}\) 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: | ||
Line 19: | Line 19: | ||
===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 \(\quad\lim\limits_{x\to\infty}{\rm f}(x)\) ? |