Integration Bee Training for Advanced #16.1 - Introduction to Limit Integration!!

Memorable Timestamp:
0:39 - A Beginner Example
2:23 - Utilize Baby Limits!!
5:42 - A Trickier L'Hopital's Rule Example
10:59 - A Tedious Example
16:56 - A Gaussian Example???
24:46 - A Nice Limit Practice
26:47 - Beware of FTC Chain Rule

[@literallydeadpool]

14:16 where does this mystery n on the outside come from? It should be x since we take d/dn?

[@literallydeadpool]

This will result in a different limit to take which is just a few more steps, I think I got x²/4+x³/6+C

[@Silver-cu5up]

oh crap 0_0, it should be x rather than n ;_;

[@onusiddartha1641]

I suggest u should also work on some real topics also
Just my suggestion bro but u r doing amazing 🔥🔥🔥

[@aidanstrong1061]

I am fairly convinced that the example at 16:56 is completely wrong. You assumed that the limit exists, that cos^n is >0, and that its valid to interchange limit and integrand. In fact, I am pretty sure the limit does not exist. Moreover, if you take the limit over even integers, it should equal infinity. The sqrt(pi/2) is equal to the limit when the integration is over (0, sqrt(n)).
Of course, I understand this is not meant to be an analysis video, but it's risky business doing these calculus tricks without justification!

[@Silver-cu5up]

Dang, unfortunately, I dont have enough knowledge to justify the proof of the convergence in that way ;_;

[@digxx]

19:25 you would have to justify by DCT.

[@Silver-cu5up]

mathematically yes (but idk too much about real analysis), but mainly DCT is always justified in integration bees unless a problem writer messed up and gave a diverging integral

[@ninadmunshi2879]

I can't tell you how long I've been waiting for this day!

[@phat5340]

I like to be anal about maths so what you need to use is the Stolz-Cesàro theorem, to do the lhopital rule you need continuity of the real funtion at that point so as to say lim x f(x) = lim n f(an) or just f(n) also I think you at least formally buthered the cos^n integral with that change of limits plus the whole bound thingy as well, I know int bees aren't that concerned about these but in collage level math competitions such integrals do appear and how you justify what ever it is that you're doing is important

[@Silver-cu5up]

Ive only heard of Stolz-Cesaro theorem, but not too familiar with it. But you are correct though that in an actual math field, justifying why we can interchange limits and integrals is important because it's not always the case (Dominating Convergence Theorem? I dont remember the topics), but of course in int bees it is assumed all integrals are convergent.