the fact, again (with a definition of e)

By l'hopital's rule, • THE FACT
The other def. of e • what is e, and the der...
I call this limit "the fact" with my students, so we can refer to this easily whenever we need to calculate such limit. ,
Limit as x goes to infinite, (1+a/x)^(bx), using definition of e,
Limit of (1+a/n)^(bn) as n goes to infinity,
Using definition of e to calculate limits,
limit of (1+1/x)^x,
www.blackpenredpen.com,
blackpenredpen,
math for fun,

[@azamingxzc9570]

I love your videos, I am not in calculus yet, precal and right now it's just algebra review, but your videos are still interesting. I don't know what derivative means or is, or integration, but I find it fun how do you do algebra and rules of those things to solve equations.

[@TrinoElrich]

Azaming Xzc you'll get there eventually, man. Fun stuff 👌

[@David-xq3bg]

Integration is basically chopping something up into infinitely small pieces and adding them together. I didn't realize this until halfway through Calc III, wish I realized this long long ago. hope it helps one day :)

[@markgraham2312]

Your two videos (lessons) on The Fact are great! Thank you!

[@blackpenredpen]

Glad to hear. Thank you.

[@rexevan6714]

I just realize that the word "blackpen" isn't even black, but you got the redpen correctly (look at your profile picture) :P

[@dannyspeagle10]

This is on of my most favorite proofs from you.

[@mangeshmalgi279]

Very lucid explanation Thanks

[@sherwan2]

Thanks for all these great videos!

[@user-wu8yq1rb9t]

Teacher ... I love your old videos too. And honestly, sometimes I love them even more than Newest one. There is a difference between your old videos and new ones. If I was on your shoes, i tried to find it because it's very very important (I know what it is!) Even when I'm in the bad mood, I watch your videos, because their change my mood ... Therefore your channel is so important to me and ... I love you teacher ❣️)

[@Sprajt]

Wow this is a great explanation of substitution! It finally makes sense to me. Thanks!

[@allisonstenz2287]

whyyyyy did the professor never mention this ??!! You saved my life thank you

[@lanasurkis8768]

your videos are very good and have a wide message thank you

[@markkennedy9767]

Hi can the definition of e also be given as the limit as x goes to minus infinity of this expression. Or is there a restriction that it has to be as x goes to plus infinity. The reason I ask is I'm trying to recall if the equivalent definition of e where x goes to zero is a one sided or two sided limit to zero. In other words, does that definition only have X going to zero from the positive side and not the negative side. Surely it's a two sided limit isn't it in the definition of e. Can you refresh my memory? Cheers

[@wonghuaixin6681]

Thank you!! your solution are so easy to understand !!!!

[@yogevsimonovich5887]

Can you do a video proving or explaining the chain rule? Love your vids (:

[@brubsi9658]

where can i find the proof of eulers number definition but without using lhopitals rule

[@bbsonjohn]

how do you prove for complex a and b

[@Cloud88Skywalker]

Hi! You take the limit of the power as the power of the limit BECAUSE "the power" is a continous function. But that's not generally the case. Negative powers are typically discontinous. Take for example x^(-1). Its discontinous, specifically at x=0. And it doesn't even have a limit there. So... does this work because it's miraculously avoiding the discontinuities?

[@angelmendez-rivera351]

Eric Ester The only place at which x^p for any real-valued p and any real-values x is discontinuous is at x = 0, provided that x^p is allowed to be complex-values. This works because we know that x is not 0 in the situation, since x = e by the limit definition of e. For all positive x, x^p is always continuous if p is real-valued.

[@khaalidmaxamuud3581]

Absolutely you explained it easily way

[@skltfz4997]

i want to know why e^1 = limit of n to inf (1+1/n)^n

[@samueldeandrade8535]

I have to talk some sh1t here. The definition e := lim (1+1/n)ⁿ is NOT just a definition. It is ESSENTIALLY THE definition of e. Because it is the original definition. I can't understand why people don't recognise that.

[@katiatzo]

Thank you Sir!

[@sinom]

Wait... Again again?

[@blackpenredpen]

Sinom yes. This is an easier way..

[@Craznar]

Easier ways always worth it. The journey is the best part :)

[@materiasacra]

It is not just another, possibly easier, way. Slowly the logical structure becomes a little more explicit: what are definitions, and what is derived from them. Early teaching is often vague on this, pointing out 'disjointed' connections to give students an intuitive grasp. Formal structure comes later.

[@guliyevshahriyar]

Thanks a lot teacher

[@dogbiscuituk]

BlackPenTransparentPen ;-)

[@steve2817]

lim x->inf (1+π/x)^ix = -1 Complekzical limit

[@GusPintarKalianGoblok]

I got e^(ab) also. YAY 😊😊😊

[@kijstanley2798]

I love this Sir

[@ivana5263]

You're an angel!

[@beezball38]

doesn't that other definition of e only converge when we approach 0 from the right?

[@angelmendez-rivera351]

Chris Beser No, it converges from the negative ray of the axis as well. This can be seen by the fact that the negative from the 1/u can be factored away, and we obtain a product (-1)^u*1/|u|, which limits with u -> 0-. Yet this means (-1)^u -> 0-, so asymptotically, it is equivalent with the limit that has u -> 0+, so the limit exists.

[@JordHaj]

6:45 blackpen.exe stopped working

[@davidseed2939]

Why do you make it so complicated, you could do this in five lines. 1) question . Y = lim x-> ( 1+ a/x) ^bx 2) definition of e using t 3) raise this equation to the power z Y = e^z 4) substitute for t, 1/t= a/x. ; Zt = bx. So z= ba 5) Y= e^ba Example a=-1/2. ; b=4. Substitute in 5). Y= e^-2

[@angelmendez-rivera351]

David Seed That is not rigorous enough. You cannot substitute backwards like that. Also, he did do it in essentially five lines.

[@aiapihud4344]

out of curiosity, if the lim x->inf (1+1/x)^x = e, then why does it equal 1 when x > 9.0072 x 10^15?

[@James2210]

Floating point errors

[@aiapihud4344]

@@James2210 what? explain in depth I've taken calc1 n that's it

[@gabe9388-g5g]

my guy

[@koenth2359]

陈露 !!

[@Whatdothenumbersmean]

mashallah

[@SavageLevel-ez1sx]

supreme :0

[@ahmednasr6171]

it still helpful

[@sunilathinagaram4441]

u r living srinivasa ramanujan

[@longgguterres-pn2jx]

How to solve lim(5n-2/5n+3)^3n

[@Hogojub]

nice