Intuition for Strange Exponential Limit

Question

I am an undergrad math student, and I am just looking for intuition or even just some reason why

$$\lim_{n\to\infty}\Big(1+\frac{x}{n}\Big)^n=e^x$$

I know you can justify the answer with L'Hospital's Rule or power series, but those don't illuminate why this would be true in a purely logical, mathematical sense.

Edit: I understand this is sort of difficult to answer and is very open ended. I think that I can clear it up a bit. Simply plugging in $n=\infty$ into the limit would, hypothetically, give $1^\infty$, which of course is an indeterminate form. So then why is this indeterminate, and what does that have to do with $e$? Or is this really just the definition of $e$?

https://math.stackexchange.com/questions/26037/intuitive-understanding-of-the-constant-e There is a good discussion around this subject in this other post — caverac, Feb 25 '21 at 22:18
It is not clear what you want. The answer is true because it can be proven. — herb steinberg, Feb 25 '21 at 22:19
@vitamind no, that's wrong. $\lim_{n\to\infty}\Big(1+\frac{1}{n}\Big)^n=e$ is the definition. Joshua Wang's answer shows how this can be derived. — miracle173, Feb 25 '21 at 22:40
@herbsteinberg that is true for all mathematical theorems. So I assume the OP he wants a proof. — miracle173, Feb 25 '21 at 22:50

Joshua Wang · Answer 1 · 2021-02-25T23:24:16.497

2

You can get this easily from the elementary definition of $e$, if that's what you want. Note:

$$\lim_{n\to\infty}\bigg(1 + \frac{x}{n}\bigg)^{n}\ \stackrel{t= \frac{n}{x}}{=}\ \lim_{t\to \infty}\bigg(\bigg(1 + \frac{1}{t}\bigg)^{t}\bigg)^{x}=\bigg(\lim_{t\to \infty}\bigg(1 + \frac{1}{t}\bigg)^{t}\bigg)^{x} = \boxed{e^{x}}$$

edited Feb 25 '21 at 23:24

answered Feb 25 '21 at 22:20

Joshua Wang

6,103

1

I find misleading the notation $\lim_{\frac{n}{x} \to \infty} \dots$ – Gabrielek Feb 25 '21 at 22:44
Why would that be? – Joshua Wang Feb 25 '21 at 22:53
1

You can use $t=x/n$ with $x$ a real constant. – Sebastiano Feb 25 '21 at 23:05
1

I agree with @sebastiano. Usually if I want to say something like that I specify somewhere that $x$ is fixed. Otherways it is a two-variables limit not well defined. – Gabrielek Feb 25 '21 at 23:20
1

@Gabrielek Thank you very much for your comment. My thanks....after :-) – Sebastiano Feb 25 '21 at 23:22
1

Edited, thank you. – Joshua Wang Feb 25 '21 at 23:24
Now...+1 ahahahah – Sebastiano Feb 25 '21 at 23:26
Wait, it is not just the substitutiom the problem. That was clear to me. The "problem" is to say somewhere that $x$ must be fixed. Otherways for $t\to \infty$ you can't tell if it is $x$ fixed and $n\to \infty$ or $n$ fixed and $x \to 0$ for example. In this contex this might be a silly comment but I think it has to be specified – Gabrielek Feb 25 '21 at 23:28

score 1 · Accepted Answer · answered Feb 25 '21 at 22:32

If you ask a mathematician what the number $e$ is and what makes it so special, he will either say it's properties in analysis, such as $$\frac{\mathrm{d}}{\mathrm{d}x} e^x = e^x,\quad \int e^x\, \mathrm{d}x = e^x+c $$ or it's beatiful combination with $i,\pi,1$ and $0$ $$e^{i\pi}+1=0,$$ the famous Euler's identity, which is usually proven by the also beatiful power series for $e$ that Euler discovered: $$\sum_{n=0}^{\infty}\frac{1}{n!}.$$

None of these formulas is the one you stated. But still, this formula has it's own history. It is the definition of $e$, how it all started, how Euler discovered further properties. Below are two interesting YouTube videos that tell you everything about the definition of $e$:

More maths (starts at 1:18)

More history (Numberfile)

Addendum: I remember quite well how I asked myself the exact same question ;).

score 1 · Answer 3 · answered Feb 25 '21 at 22:39

You just have the show the limit of the natural log of the expression is $x$.

Indeed, from a high school limit, $$\ln\biggl(1+\frac xn\biggr)^{\!n}=n\ln\Bigl(1+\frac xn\Bigr)=x\,\underbrace{\frac{\ln\Bigl(1+\frac xn\Bigr)}{\frac xn}}_{\substack{\downarrow \rlap{\:n\to\infty}\\1}} $$ tends to $x$ as $n$ tends to $\infty$.

user2661923 · Answer 4 · 2021-02-25T22:43:32.700

0

Alternative intuition

$e$ is the real number such that $\frac{d}{dx} e^x = e^x.$

$\frac{d}{dx} (1 + [x/n])^n = n \times (1 + [x/n])^{(n-1)} \times [1/n] = (1 + [x/n])^{(n-1)}.$

By binomial expansion: $\lim_{n\to\infty} (1 + [x/n])^n ~=~ \lim_{n\to\infty} (1 + [x/(n+1)])^n~=~ \lim_{n\to\infty} (1 + [x/n])^{(n-1)}.$

edited Feb 25 '21 at 22:43

answered Feb 25 '21 at 22:38

user2661923

35,619
3
17
39

Intuition for Strange Exponential Limit

4 Answers4