Linked Questions

20
votes
3answers
143k views

Limit of $(1+ x/n)^n$ when $n$ tends to infinity [duplicate]

Does anyone know the exact proof of this limit result? $$\lim_{n\to\infty} \left(1+\frac{x}{n}\right)^n = e^x$$
3
votes
6answers
446 views

Showing that $\lim_{n \rightarrow \infty} \left( 1 + \frac{r}{n} \right)^{n} = e^{r} $. [duplicate]

I know that $ \displaystyle \lim_{n \rightarrow \infty} \left( 1 + \frac{1}{n} \right)^{n} = e $, but how do I show that $$ \lim_{n \rightarrow \infty} \left( 1 + \frac{r}{n} \right)^{n} = e^{r}? $$ ...
2
votes
1answer
303 views

Different proofs of $\lim_{x\to \infty}\left(1+ \frac{1}{n}\right)^n =e$ [duplicate]

I recently was teaching my friend about the number $e$. I introduced him the number by using the compound interest thing . Then I wrote down the general result -$$\lim_{x\to \infty}\left(1+ \frac{1}{...
1
vote
4answers
124 views

Finding the limit of the sequence $x(n) = (1+2/n)^n$ [duplicate]

What we are allowed to use - 1) The fact that limit of $(1+1/n)^n$ exists and assumed to be some real number $e$ 2) Subsequencial properties of limits of sequences 3) Basic properties of limit In the ...
0
votes
2answers
176 views

How to show that $\lim_{n\rightarrow \infty }(1+\frac{x}{n})^{n}=e^{x}$ [duplicate]

I have a problem: $E(x)=e^x$, $L(x)=\ln (x)$, $E^{-1}(x)=L(x)$. Show that $\lim_{n\rightarrow \infty }(1+\frac{x}{n})^{n}=e^{x}$ Hint: use $f(t)=\ln (1+xt)$ and look at $f'(0), x\neq 0$. I ...
0
votes
0answers
138 views

$e^x$ as a limit proof [duplicate]

So I am currently going over some old calculus concepts and ran cross the exponential function expressed as a limit: $$e^x = \lim_{n\rightarrow \infty}\left(\!1+\frac{x}{n}\!\right)^n $$ Where ...
-1
votes
1answer
99 views

Continuous compounding problem [duplicate]

How long does it take an investment to double if continuously compounded at a 6% rate? Is this right? \begin{align*} e^{0.06 \cdot t} & = 2\\ 0.06t & = \ln(2)\\ t & = \frac{\ln 2}{0.06}\\ ...
0
votes
1answer
76 views

Proving the limit $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n=e$ [duplicate]

I want to prove that $\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n=e$ There is a solution of the sum provided in my text book. There the expansion of $(1+\frac{1}{n})^n$ is like below: $(1+\frac{1}{n})...
1
vote
2answers
57 views

Determining the convergence and divergence [duplicate]

So I have a problem of $$a_n = \left(1 + \frac{2}{n}\right)^n$$ I need to determine whether it is diverging or converging and find the limit if it is converging I found an answer on symbol lab of $...
-2
votes
1answer
53 views

Basic evaluation of a limit [duplicate]

How would you evaluate the following limit as n goes to infinity? $$\lim \frac {1}{(1+\frac {1}{n})^n}$$ I would of thought that this would evaluated to be, $$\lim \frac {1}{(1)^n} = 0 $$ However ...
0
votes
0answers
34 views

How to compute this limit with different operations of $x$? [duplicate]

In another answer I saw, there is this expression $$\lim _{ x\rightarrow \infty }{ x{ \left( 1-\frac { 1 }{ x } \right) }^{ x } } =\lim _{ x\rightarrow \infty }{ \frac { x }{ e } } =\infty$$ ...
0
votes
0answers
12 views

problem on translation in quantum mechanics [duplicate]

$$\lim_{N\to \infty} \left(1-\frac{x}{N}\right)^N=\exp(-x)$$ Can anyone give a proof for it, I see a similar result in quantum mechanics ( Momentum as a generator of translation ). The textbook does ...
35
votes
14answers
2k views

Intuitive proofs that $\lim\limits_{n\to\infty}\left(1+\frac xn\right)^n=e^x$

At this link someone asked how to prove rigorously that $$ \lim_{n\to\infty}\left(1+\frac xn\right)^n = e^x. $$ What good intuitive arguments exist for this statement? Later edit: . .&...
21
votes
9answers
2k views

Finding the limit of $\left(\frac{n}{n+1}\right)^n$

Find the limit of: $$\lim_{n\to\infty}\left(\frac{n}{n+1}\right)^n$$ I'm pretty sure it goes to zero since $(n+1)^n > n^n$ but when I input large numbers it goes to $0.36$. Also, when factoring: $...
9
votes
7answers
975 views

How to evaluate the limit where something is raised to a power of $x$?

I am attempting to evaluate the following limit: $$\lim_{x\to \infty} \Biggl(\frac{x+3}{x+8}\Biggl)^x$$ I was wondering if anyone could share some strategies for evaluating limits raised to a power ...

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