1. ## limit of (1+1/x)^x

how do we show the limit of (1+1/x)^x is e (as x tends to infinity) by bounding it above and showing that it is an increasing sequence?
Thanks for any help

2. See natural log base and the links on that page on PlanetMath. They consider a sequence $\displaystyle (1+1/n)^n$, not a function, however.

3. Originally Posted by hmmmm
how do we show the limit of (1+1/x)^x is e (as x tends to infinity) by bounding it above and showing that it is an increasing sequence?
Thanks for any help
Let's consider the function $\displaystyle \lambda(x) = \ln \{(1 + \frac{1}{x})^{x}\}= x\ \ln (1+\frac{1}{x})$. For $\displaystyle |x|>1$ is...

$\displaystyle \displaystyle \lambda (x) = 1 - \frac{1}{2\ x} + \frac{1}{3\ x^{2}} - ...$ (1)

... so that is...

$\displaystyle \displaystyle \lim_{x \rightarrow \infty} \lambda(x) = 1 \implies \lim_{x \rightarrow \infty} (1+\frac{1}{x})^{x}= e$ (2)

From (1) it is easy to derive that if $\displaystyle x_{2}>x_{1}>1$ is $\displaystyle \lambda(x_{2}) > \lambda(x_{1})$ so that $\displaystyle \lambda (x)$ is an increasing function...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$