math104-s21:s:johndufek
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math104-s21:s:johndufek [2021/05/07 19:10] 73.71.56.149 |
math104-s21:s:johndufek [2026/02/21 14:41] (current) |
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| //Example: say you want to prove the limit of $\frac{4n^3 + 3n}{n^3 - 6} => \frac{3n+24}{n^3 - 6} = 4 $ . It would be hard to isolate $n$ to find such an $N > 0$. \\ | //Example: say you want to prove the limit of $\frac{4n^3 + 3n}{n^3 - 6} => \frac{3n+24}{n^3 - 6} = 4 $ . It would be hard to isolate $n$ to find such an $N > 0$. \\ | ||
| Hence bound the sequence by something easier i.e. $3n + 24 \leq 27n$. Thus $\frac{27n}{n^3 - 6} < \epsilon$. Building off this technique we know the denominator $n^3 - 6 \geq \frac{n^3}{2}$ . Hence $\frac{27n}{n^3 / 2} < \epsilon => \frac{54}{n^2} < \epsilon$ .Then from here you can easily find an $N > 0$ s.t. the limit definition holds. This technique is displayed in Ross page 40 - 41.// \\ | Hence bound the sequence by something easier i.e. $3n + 24 \leq 27n$. Thus $\frac{27n}{n^3 - 6} < \epsilon$. Building off this technique we know the denominator $n^3 - 6 \geq \frac{n^3}{2}$ . Hence $\frac{27n}{n^3 / 2} < \epsilon => \frac{54}{n^2} < \epsilon$ .Then from here you can easily find an $N > 0$ s.t. the limit definition holds. This technique is displayed in Ross page 40 - 41.// \\ | ||
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| + | //See Ross textbook chapter 9 for useful limit properties and theorems.// | ||
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| + | **Sequences**\\ | ||
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| $(b)\ d(p,q)\ =\ d(q,p)$ \\ | $(b)\ d(p,q)\ =\ d(q,p)$ \\ | ||
| $( c )\ d(p,q)\ \le \ d(p,r)\ +\ d(r,q),\ for\ any\ r\ \in \ X$ \\ | $( c )\ d(p,q)\ \le \ d(p,r)\ +\ d(r,q),\ for\ any\ r\ \in \ X$ \\ | ||
| - | $d(x,y)\ =\ |x-y|\ (x,y\ \in \ \{R}^k \ )$ \\ | + | $d(x,y)\ =\ |x-y|\ (x,y\ \in \ \R^k \ )$ \\ |
| . A K-cell is convex and compact\\ | . A K-cell is convex and compact\\ | ||
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| 2) | 2) | ||
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| + | **Limits**\\ | ||
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| + | 1) How can we know/ differentiate certain approaches to particular limits of sequences that satisfy L' | ||
math104-s21/s/johndufek.1620414629.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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