1. ## Prove that lim...=e

Prove that

$\lim_{h \to 0}(1+h)^{1/h}=e$

Hints outlined for this question are as follows.

Since $\frac{d}{dx}\ln1=1$,
As $h \to 0$, $\frac{\ln(1+h)-\ln1}{h}=\frac{\ln(1+h)}{h} \to 1$
A second hint is that if $g$ is continuous at $c$ and $f$ is continuous at $g(c)$, then $f\circ g$ is continuous at $c$.

You are not allowed to use L'Hospital's Rule to solve this question. This is one of the requirements, I'm afraid.

2. Do you understand that $\lim _{n \to \infty } \left( {1 + \frac{1}{n}} \right)^n = e$?
If so let $h=\frac{1}{n}$.

3. Originally Posted by Plato
Do you understand that $\lim _{n \to \infty } \left( {1 + \frac{1}{n}} \right)^n = e$?
If so let $h=\frac{1}{n}$.
Didn't quite get what you were getting at at first, but then I realized it. Thanks.

4. Originally Posted by Runty
Prove that

$\lim_{h \to 0}(1+h)^{1/h}=e$

Hints outlined for this question are as follows.

Since $\frac{d}{dx}\ln1=1$,
As $h \to 0$, $\frac{\ln(1+h)-\ln1}{h}=\frac{\ln(1+h)}{h} \to 1$
A second hint is that if $g$ is continuous at $c$ and $f$ is continuous at $g(c)$, then $f\circ g$ is continuous at $c$.

You are not allowed to use L'Hospital's Rule to solve this question. This is one of the requirements, I'm afraid.
Let $1=\lim_{x\to0}\frac{\ln(1+x)}{x}=\lim_{x\to0}\ln\l eft(\left(1+x\right)^{\frac{1}{x}}\right)=\ln\left (\lim_{x\to0}\left(1+x\right)^{\frac{1}{x}}\right)$. The last part follows from the hint stuff. The conclusion follows by exponentiation.