# lim (1+1/x)^x

• Apr 21st 2013, 05:06 AM
kicma
lim (1+1/x)^x
I need a proof that lim(1+1/x)^x=e.
We defined that e=lim(1+1/n)^n and I need to prove that it's same for x (real number).
Thanks.
• Apr 21st 2013, 05:39 AM
Re: lim (1+1/x)^x
The question seems trivial, as:

$\displaystyle \lim_{n\rightarrow \infty }\left ( 1+\frac{1}{n} \right )^{n}=e$

Is the exact same statement as:

$\displaystyle \lim_{x\rightarrow \infty }\left ( 1+\frac{1}{x} \right )^{x}=e$

Just let n = x.
$\displaystyle n \epsilon \mathbb{R}$
$\displaystyle x \epsilon \mathbb{R}$

Though, this also seems trivial.. Even if they were integers or natural numbers, it would still approach the value of 'e' as the limit went to infinity.

Have you done the proof for the 'n' in class?
• Apr 21st 2013, 06:14 AM
Plato
Re: lim (1+1/x)^x
Quote:

The question seems trivial, as:
$\displaystyle \lim_{n\rightarrow \infty }\left ( 1+\frac{1}{n} \right )^{n}=e$
Is the exact same statement as:
$\displaystyle \lim_{x\rightarrow \infty }\left ( 1+\frac{1}{x} \right )^{x}=e$
Just let n = x.
$\displaystyle n \epsilon \mathbb{R}$
$\displaystyle x \epsilon \mathbb{R}$
Though, this also seems trivial.. Even if they were integers or natural numbers, it would still approach the value of 'e' as the limit went to infinity.

That is no proof. Real numbers are not all integers.

To do this we need to have two additional facts: assume that $\displaystyle x\ge 1$ then
$\displaystyle \\1)~\lim_{n\rightarrow \infty }\left ( 1+\frac{1}{n+1} \right )^{n}=e\\\\2)~\lim_{n\rightarrow \infty }\left ( 1+\frac{1}{n} \right )^{n+1}=e$

Now use the floor function (greatest integer function): $\displaystyle \left\lfloor x \right\rfloor \leqslant x < \left\lfloor x \right\rfloor + 1$.

Thus $\displaystyle \left( {1 + \frac{1}{{\left\lfloor x \right\rfloor }}} \right) \geqslant \left( {1 + \frac{1}{x}} \right) \geqslant \left( {1 + \frac{1}{{\left\lfloor x \right\rfloor + 1}}} \right)$

So $\displaystyle {\left( {1 + \frac{1}{{\left\lfloor x \right\rfloor }}} \right)^{\left\lfloor x \right\rfloor + 1}} \geqslant {\left( {1 + \frac{1}{x}} \right)^x} \geqslant {\left( {1 + \frac{1}{{\left\lfloor x \right\rfloor + 1}}} \right)^{\left\lfloor x \right\rfloor }}$
• Apr 21st 2013, 07:07 AM
x3bnm
Re: lim (1+1/x)^x
We know that:

$\displaystyle \lim_{n\rightarrow \infty }\left ( 1+\frac{1}{n} \right )^{n} = \lim_{n\rightarrow \infty } e^{\ln{\left( 1+\frac{1}{n} \right)^{n}}}$

$\displaystyle \text{Let } N = \frac{1}{n}$

and as $\displaystyle n \rightarrow \infty$ so $\displaystyle N \rightarrow 0$

$\displaystyle \therefore \lim_{n\rightarrow \infty } e^{\ln{\left( 1+\frac{1}{n} \right)^{n}}} = \lim_{N\rightarrow 0 } e^{\ln{\left( 1+N \right)^{\frac{1}{N}}}}$

\displaystyle \begin{align*}\lim_{N \rightarrow 0 } e^{\ln{(1+N)^{1/N}}} =& \lim_{N \rightarrow 0 } e^{\frac{\ln{(1+N)}}{N}}\\=& e^{\lim_{N \rightarrow 0} \frac{\ln{(1+N)}}{N}}.......[\text{by power rule of limit}]\end{align*}

Using I'Hopital's rule:

$\displaystyle \lim_{N \rightarrow 0} \frac{\ln{(1+N)}}{N} = \lim_{N \rightarrow 0} \frac{\frac{1}{1+N}}{1} = 1$

$\displaystyle \lim_{n\rightarrow \infty }\left ( 1+\frac{1}{n} \right )^{n} = e^{1} = e$

$\displaystyle \text{Q.E.D}$
• Apr 21st 2013, 07:20 AM
Plato
Re: lim (1+1/x)^x
Quote:

Originally Posted by x3bnm
$\displaystyle \lim_{n\rightarrow \infty }\left ( 1+\frac{1}{n} \right )^{n} = e^{1} = e$

@x3bnm, Did you read the OP?
Had you, you would have noted that the fact that $\displaystyle \lim_{n\rightarrow \infty }\left ( 1+\frac{1}{n} \right )^{n}=e$ was never in question.
In fact, the OP stipulates that is the definition of $\displaystyle e$.

The you are asked to show that $\displaystyle {\lim _{x \to \infty }}{\left( {1 + \frac{1}{x}} \right)^x} = e$.
• Apr 21st 2013, 07:23 AM
x3bnm
Re: lim (1+1/x)^x
Quote:

Originally Posted by Plato
@x3bnm, Did you read the OP?
Had you, you would have noted that the fact that $\displaystyle \lim_{n\rightarrow \infty }\left ( 1+\frac{1}{n} \right )^{n}=e$ was never in question.
In fact, the OP stipulates that is the definition of $\displaystyle e$.

The you are asked to show that $\displaystyle {\lim _{x \to \infty }}{\left( {1 + \frac{1}{x}} \right)^x} = e$.

Sorry Plato. My bad. I thought(erroneously) he wanted to know why it's equal to $\displaystyle e$. Readers...please disregard my last post. Sorry.