# “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,

www.blackpenredpen.com

source

Wolfram alpha https://www.wolframalpha.com/input/?i=limit+1%5Ex+as+x+goes+to+infinity

LOL

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

And how proof the L'Hospital's Rule ?

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?

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

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)

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

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.

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

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

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.

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

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

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

you're just A-OK!

Thanx a lot from Paris!!

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

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.

I love your videos What is your name?

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?

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

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.

I like this channel. It states facts.

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

Cannot read the red bits.

Seriesly 😂

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

lim (1+i/n)^(πn) = -1

ï love you man thank you

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

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?

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!

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. 🙁

hi

i love your channel ! i see all of your videos!

lets do math for fun

bye

… that's THE FACT Jack … !

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

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

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

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

heavy duty !

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))

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!

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

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)

Hey … Can you do some number theory problems ?

You never fail to impress me!

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

Thanks! Really good explanation!

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