10 Challenging Limit Problems! Can you survive?😱(part1)🔎Riemann Sum, Squeeze Theorem, Floor Function

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#calculus #limit #riemannsum #squeezetheorem #math #apcalculus #midterm #exam

[@user-jq5us6di2p]

影片做的活潑有創意，謝謝老師的分享，希望大家記得按讚多多支持喔！

[@ThisisChuan]

感謝支持❤️

[@leo-ec5xu]

影片做得很棒👍

[@ThisisChuan]

感謝支持❤️

[@goldfist8711]

Great video!

[@ThisisChuan]

Thank you! ❤️

[@brightmoonchiefyue]

第三題過程這樣寫的話，閱卷老師可以直接給零分。floor function 不是連續函數，極限怎麼可以直接搬進去？

[@user-qd9fo4gl9y]

第一題直接開羅，第二題用夾，第三題先配方，第四題原本想說直接開羅，原來要先變成e，第五題用黎曼

[@MH-sf6jz]

第二題為什麼 {(1+3/x)^(x/3)}^(3(x-1)/x) 可以把base和exponent 分開take limit？

[@MH-sf6jz]

第三題直接把limit pass進去floor function 有點misleading，9減去那個limit應該還是9，所以找你的寫法那個floor function的結果（如果pass limit進去）應該還是9。更加準確的說法應該要考慮x在一個small enough neighborhood of 3， 然後bound 9-(x-3)^2 above strictly by 9然後 再lower bound by 8，這樣才能conclude floor function 是8吧。

[@ThisisChuan]

感謝提醒🙏
計算題這樣寫確實不大嚴謹，我要表達的是可以初步用判斷得到答案是 8。

[@choke7273]

For the 5th question, I think it is possible to just integrate the top and bottom, as the error becomes infinitely small as n approaches infinity, so you will obtain (n^(3)/3)/((2n^(3/2)/3))^2 = 3/4.
The reason why the error will be very small is that when you draw the function x^2 and x^1/2, they will both go to infinity, and the error of the calculation is just the difference of area between the function and the function where x is an integer, it is hard to explain with words, but you will understand if you draw the graph yourself

[@choke7273]

For example, 1+2+3+...n=(n+1)n/2, and integral x dx = x²/2, well if you limit n to infinity it becomes n²/2 just like the integral

[@choke7273]

But I would admit your method is the bigger brain one

[@ThisisChuan]

This idea is good. More precisely, simply divide both the numerator and denominator by n^3, and you'll obtain the Riemann sum in both the numerator and denominator.
( ∫ x² dx )/( ∫ x½ dx )² from 0 to 1
= (1/3)/(2/3)² = (1/3)/(4/9) = 3/4.

[@user-qx6qg6wr3v]

我以為第四題又要拋棄羅畢達了xd 難得看到羅畢達 第五題感覺是在考對summation 的基本概念夠不夠強跟熟練

[@ThisisChuan]

羅畢達還是會給鏡頭的 XD

[@user-ks9lv9jg9c]

用遊戲王的音樂來解題 必須給讚

[@ThisisChuan]

決鬥⚔️

[@user-yp7ki3py5m]

窩似被遊戲王吸引來的😂

[@ThisisChuan]

決鬥😆

[@R_Chuang]

你第一題懂夾擠定理，但後面卻說趨近極限忽略不計，這很奇怪，好像懂數學又不懂