“THE FACT”



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),
Limit of (1+a/n)^(bn) as n goes to infinity,
Use l’hopital’s rule to calculate limit,
l’hopital’s rule examples,

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47 thoughts on ““THE FACT”

  1. Victor Lacerda September 25, 2017 at 7:19 am

    Hi, it's possible proof "The Fact" without the L'Hospital's Rule ?.

    And how proof the L'Hospital's Rule ?

  2. Samuel Andrade September 25, 2017 at 7:19 am

    I see a problem there. You use the derivative of ln to proof The Fact, and use The Fact to get the derivative of ln?

  3. travisbaskerfield September 25, 2017 at 7:19 am

    Put = ay and the result follows at once. I guess he wanted to show the principle.

  4. Robert Richards September 25, 2017 at 7:19 am

    I thought I recognized the function. After you got the answer, I noticed this is what we called the "Pert" function for continuously compounding yearly interest.

    V = P(1 + R/X)^(T*X)

    P=initial principal
    R=interest rate
    T=number of years the money is in the bank
    X=the number of times a year you compound the interest (1 for yearly, 12 for monthly, 365 for daily, etc)

    Compounded continuously, X=infinity, so V = P*e^(R*T)

  5. Adam McMillan September 25, 2017 at 7:19 am

    Could the process not be simplified by defining a new variable "n" which is equal to x/a and acknowledging that as x approaches infinity so does n and from there rewriting the expression as
    (1+1/n)^abn
    which is equal to
    ( (1+1/n)^n )^ab
    and since we know that the limit of (1+1/n)^n as n approaches infinity is e, we can rewrite this as
    (e)^ab

  6. Richard Farrer September 25, 2017 at 7:19 am

    I enjoy your videos but I'm slightly worried by this one as it could mislead people. You haven't shown the limit is e^(ab). Rather, you have shown that if there is a limit then it must be e^(ab). I would have liked you to have mentioned why the limit must exist.

  7. Vfdking September 25, 2017 at 7:19 am

    If I used log base 10 instead of ln I would have a different answer. Why does it have to be ln

  8. brian554xx September 25, 2017 at 7:19 am

    bprp, you are awesome, and you are a teacher. Can you teach me (or whoever) to be awesome?

  9. Chen Yi Lee September 25, 2017 at 7:19 am

    The expression can be rearranged to Limit of [((1+a/x)^(x/a))]^(ab) when x approach infinity. The expression in [ ..] is the definition of exp(1) , Therefore, your expression is Exp(ab) No need to do anything else.

  10. Βασίλης Μουζάκης September 25, 2017 at 7:19 am

    How do we know L is positive in order to use ln in it?

  11. thisisaproaccount September 25, 2017 at 7:19 am

    why could you take the derivative of the numerator and denominator? do you have a video explaining this (or another good source)?

  12. saitaro September 25, 2017 at 7:19 am

    The cool accent, mic and apparent joy from calculus make this video. As all others from this guy. Goddamn do I subscribe!

  13. Claude Barthelemy September 25, 2017 at 7:19 am

    you're just A-OK!

    Thanx a lot from Paris!!

  14. Petter Houting September 25, 2017 at 7:19 am

    find a and b for lim x->inf (1+a/x)^bx = pi

  15. SpiderWick12 September 25, 2017 at 7:19 am

    Could someone explain why f(lim g(x)) only equals lim(f(g(x)) when f(x) is continuous? in this case, f(x) = lnx.

  16. Curie Novak September 25, 2017 at 7:19 am

    I love your videos What is your name?

  17. Chvocht - September 25, 2017 at 7:19 am

    You don't have to use a natural log for the calculation though, right? Would you get to the same result if you used a normal logarithm?

  18. Canard Paa September 25, 2017 at 7:19 am

    Simpler
    L = lim x->∞ (1+a/x)^bx
    sub x = ay
    L = lim y->∞ (1+1/y)^aby
    L^*(1/ab) = lim y->∞ ((1+1/y)^y)ab)^(1/ab)
    L^*(1/ab) = lim y->∞ (1+1/y)^y = e
    L = e^ab
    lim x->∞ (1+a/x)^bx = e^ab

  19. Java Juicer September 25, 2017 at 7:19 am

    I still don't understand why ln(lim(f(x))) = lim(ln(f(x))). You said that ln was a continuous function, which I understand, but I don't understand how that yields the swap correct algebraically.

  20. WarpRulez September 25, 2017 at 7:19 am

    I like this channel. It states facts.

  21. Filmon Tewolde September 25, 2017 at 7:19 am

    Question How is any of this going to make you money? It is crazy how much money and time school waste.

  22. Gwilym ap Iorwerth September 25, 2017 at 7:19 am

    Cannot read the red bits.

  23. Kenan Kenobe September 25, 2017 at 7:19 am

    Seriesly 😂

  24. RB September 25, 2017 at 7:19 am

    I don't recall much but I don't think you're able to assume the limit is equal to 'L' without showing the limit exists in the first place

  25. roglo September 25, 2017 at 7:19 am

    lim (1+i/n)^(πn) = -1

  26. han wadou September 25, 2017 at 7:19 am

    ï love you man thank you

  27. Daniel Titchener September 25, 2017 at 7:19 am

    Setting a, = b = 1 is a nice way of getting to compound interest tending to e

  28. Cherrie Pie September 25, 2017 at 7:19 am

    just wondering if I replaced the natural log with other logs wouldnt that change 'The Fact'? like if I use lg it would end up 10^ab, no?

  29. PlasmaCrab _ September 25, 2017 at 7:19 am

    I'm currently taking Precalculus in High School, but the way you explained it somehow makes me understand it! The videos you've been making are awesome, and I hope you continue showing us things as cool as this!

  30. stardestroyer19 September 25, 2017 at 7:19 am

    Man, I really wish I'd seen your videos during A levels. You are so fun in your explenations. Now I'm studying mathematics at uni and while your vids are great I don't get the same satisfaction due to the fact that I can tell the answers myself quite easily. 🙁

  31. Dario Fervenza September 25, 2017 at 7:19 am

    hi
    i love your channel ! i see all of your videos!
    lets do math for fun
    bye

  32. WJL September 25, 2017 at 7:19 am

    … that's THE FACT Jack … !

  33. bonxbonx September 25, 2017 at 7:19 am

    What if you take log base 10 instead of ln? would that change the answer?

  34. Greg Brown September 25, 2017 at 7:19 am

    x=(x^-1)^-1=1/1/x

  35. Vecht September 25, 2017 at 7:19 am

    That's cheating! Do it without logarithms or exponents!

  36. João Viana September 25, 2017 at 7:19 am

    Isn't it cheating that you used ln to calculate e? Since ln is based on e

  37. realcygnus September 25, 2017 at 7:19 am

    heavy duty !

  38. luca nina September 25, 2017 at 7:19 am

    Once I found a similar problem with a more variable in the denominator.

    lim x->infinity ((x+a)/(x+b))^cx

    following the same steps of the video

    -log of both sides
    -arranging to a fraction indeterminate form
    – de hospital rule

    it came out

    e^(c(a-b))

  39. Joshua Garcia September 25, 2017 at 7:19 am

    Yes, that's very fine. Just keep posting em and me and my mates'll watch them. We're very entertained. Thanks so much for these videos. You're way fun and captivating than my professors!

  40. Iamyou iamyou September 25, 2017 at 7:19 am

    Just discovered your channel from e^pi and pi^e video , and absolutely love your content

  41. MyBigRed September 25, 2017 at 7:19 am

    The answer if obvious if you recognize this as the limit of the compound interest formula as the compounding period goes to infinity. P(1+r/n)^(t*n) – > Pe^(rt)

  42. saikat93ify September 25, 2017 at 7:19 am

    Hey … Can you do some number theory problems ?

  43. Eleazar Almazan September 25, 2017 at 7:19 am

    You never fail to impress me!

  44. FakeName 123 September 25, 2017 at 7:19 am

    I really want to insult you for choosing such a cumbersome microphone, but the sound quality speaks for itself.

  45. weerman44 September 25, 2017 at 7:19 am

    Thanks! Really good explanation!

  46. Marian P. Gajda September 25, 2017 at 7:19 am

    I really love your enthusiasm. Thank you for your videos and the magnificent lessons!

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