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I was wondering how we could use the limit definition

$$ \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$

to find the derivative of $e^x$, I get to a point where I do not know how to simplify the indeterminate $\frac{0}{0}$. Below is what I have already done

$$\begin{align} &\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\ &\lim_{h \rightarrow 0} \frac{e^{x+h}-e^x}{h} \\ &\lim_{h \rightarrow 0} \frac{e^x (e^h-1)}{h} \\ &e^x \cdot \lim_{h \rightarrow 0} \frac{e^h-1}{h} \end{align}$$

Where can I go from here? Because, the $\lim$ portion reduces to indeterminate when $0$ is subbed into $h$.

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How do you define $e$? –  Adam Saltz Apr 12 '13 at 2:22
    
@AdamSaltz $e$ as in euler's constant? –  gekkostate Apr 12 '13 at 2:23
    
@julien Sorry, I don't understand what you mean by "the function of $e^x$". Can you please clarify? –  gekkostate Apr 12 '13 at 2:24
    
@julien I don't think I understand what it means to define a function. Do you mean if it is an exponential function or a one to one function? –  gekkostate Apr 12 '13 at 2:27
    
Some people define the exponential $e^x$ as the unique solution of the ode: $y'=y$ and $y(0)=1$. For these people, no needto prove that $\frac{d}{dx}e^x=e^x$. It holds by definition. Now given your title, there are I think two possibilities: $e^x=\lim_n \sum_{k=0}^n\frac{x^k}{k!}$, or $e^x=\lim_n\left(1+\frac{x}{n}\right)^n$. –  1015 Apr 12 '13 at 2:36
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5 Answers

up vote 11 down vote accepted

Sometimes one defines $e$ as the (unique) number for which $$\tag 1 \lim_{h\to 0}\frac{e^h-1}{h}=1$$

In fact, there are two possible directions.

$(i)$ Start with the logarithm. You'll find out it is continuous monotone increasing on $\Bbb R_{>0}$, and it's range is $\Bbb R$. It follows $\log x=1$ for some $x$. We define this (unique) $x$ to be $e$. Some elementary properties will pop up, and one will be $$\tag 2 \lim\limits_{x\to 0}\frac{\log(1+x)}{x}=1$$

Upon defining $\exp x$ as the inverse of the logarithm, and after some rules, we will get to defining exponentiation of $a>0\in \Bbb R$ as $$a^x:=\exp(x\log a)$$

In said case, $e^x=\exp(x)$, as we expected. $(1)$ will then be an immediate consequence of $(2)$.

$(ii)$ We might define $$e=\sum_{k=0}^\infty \frac 1 {k!}$$ (or the equivalent Bernoulli limit). Then, we may define $$\exp x=\sum_{k=0}^\infty \frac{x^k}{k!}$$ Note $$\tag 3 \exp 1=e$$

We define the $\log$ as the inverse of the exponential function. We may derive certain properties of $\exp x$. The most important ones would be $$\exp(x+y)=\exp x\exp y$$ $$\exp'=\exp$$ $$\exp 0 =1$$

In particular, we have that $\log e=1$ by. We might then define general exponentiation yet again by $$a^x:=\exp(x\log a)$$

Note then that again $e^x=\exp x$. We can prove $(1)$ easily recurring to the series expansion we used.


ADD As for the definition of the logarithm, there are a few ones. One is $$\log x=\int_1^x \frac{dt}{t}$$

Having defined exponentiation of real numbers using rationals by $$a^x=\sup\{a^r:r\in\Bbb Q\wedge r<x\}$$

we might also define $$\log x=\lim_{k\to 0}\frac{x^k-1}{k}$$

In any case, you should be able to prove that

$$\tag 1 \log xy = \log x +\log y $$ $$\tag 2 \log x^a = a\log x $$ $$\tag 3 1-\dfrac 1 x\leq\log x \leq x-1 $$ $$\tag 4\lim\limits_{x\to 0}\dfrac{\log(1+x)}{x}=1 $$ $$\tag 5\dfrac{d}{dx}\log x = \dfrac 1 x$$

What you want is a direct consequence of either $(4)$ or $(5)$, or of the first sentence in my post.


ADD We can prove that for $x \geq 0$ $$\lim\left(1+\frac xn\right)^n=\exp x$$ from definition $(ii)$.

First, note that $${n\choose k}\frac 1{n^k}=\frac{1}{{k!}}\frac{{n\left( {n - 1} \right) \cdots \left( {n - k + 1} \right)}}{{{n^k}}} = \frac{1}{{k!}}\left( {1 - \frac{1}{n}} \right)\left( {1 - \frac{2}{n}} \right) \cdots \left( {1 - \frac{{k - 1}}{n}} \right)$$

Since all the factors to the rightmost are $\leq 1$, we can claim $${n\choose k}\frac{1}{{{n^k}}} \leqslant \frac{1}{{k!}}$$

It follows that $${\left( {1 + \frac{x}{n}} \right)^n}=\sum\limits_{k = 0}^n {{n\choose k}\frac{{{x^k}}}{{{n^k}}}} \leqslant \sum\limits_{k = 0}^n {\frac{{{x^k}}}{{k!}}} $$

It follows that if the limit on the left exists, $$\lim {\left( {1 + \frac{x}{n}} \right)^n} \leqslant \lim \sum\limits_{k = 0}^n {\frac{{{x^k}}}{{k!}}} = \exp x$$

Note that the sums in $$\sum\limits_{k = 0}^n {{n\choose k}\frac{{{x^k}}}{{{n^k}}}} $$

are always increasing, which means that for $m\leq n$

$$\sum\limits_{k = 0}^m {{n\choose k}\frac{{{x^k}}}{{{n^k}}}}\leq \sum\limits_{k = 0}^n {{n\choose k}\frac{{{x^k}}}{{{n^k}}}}$$

By letting $n\to\infty$, since $m$ is fixed on the left side, and $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{k!}}\left( {1 - \frac{1}{n}} \right)\left( {1 - \frac{2}{n}} \right) \cdots \left( {1 - \frac{{k - 1}}{n}} \right) = \frac{1}{{k!}}$$

we see that if the limit exists, then for each $m$, we have $$\sum\limits_{k = 0}^m {\frac{{{x^k}}}{{k!}}} \leqslant \lim {\left( {1 + \frac{x}{n}} \right)^n}$$

But then, taking $m\to\infty$ $$\exp x = \mathop {\lim }\limits_{m \to \infty } \sum\limits_{k = 0}^m {\frac{{{x^k}}}{{k!}}} \leqslant \lim {\left( {1 + \frac{x}{n}} \right)^n}$$

It follows that if the limit exists $$\eqalign{ & \exp x \leqslant \lim {\left( {1 + \frac{x}{n}} \right)^n} \cr & \exp x \geqslant \lim {\left( {1 + \frac{x}{n}} \right)^n} \cr} $$ which means $$\exp x = \lim {\left( {1 + \frac{x}{n}} \right)^n}$$ Can you show the limit exists?

The case $x<0$ follows now from $$\displaylines{ {\left( {1 - \frac{x}{n}} \right)^{ - n}} = {\left( {\frac{n}{{n - x}}} \right)^n} \cr = {\left( {\frac{{n - x + x}}{{n - x}}} \right)^n} \cr = {\left( {1 + \frac{x}{{n - x}}} \right)^n} \cr} $$

using the squeeze theorem with $\lfloor n-x\rfloor$, $\lceil n-x\rceil$, and the fact $x\to x^{-1}$ is continuous. We care only for terms $n>\lfloor x\rfloor$ to make the above meaningful.

NOTE If you're acquainted with $\limsup$ and $\liminf$; the above can be put differently as $$\eqalign{ & \exp x \leqslant \lim \inf {\left( {1 + \frac{x}{n}} \right)^n} \cr & \exp x \geqslant \lim \sup {\left( {1 + \frac{x}{n}} \right)^n} \cr} $$ which means $$\lim \inf {\left( {1 + \frac{x}{n}} \right)^n} = \lim \sup {\left( {1 + \frac{x}{n}} \right)^n}$$ and proves the limit exists and is equal to $\exp x$.

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What is $\text{exp}$? and $\text{sup}$? –  gekkostate Apr 12 '13 at 2:47
    
What a job, Peter, +1. It remains $\lim (1+x/n)^n$... –  1015 Apr 12 '13 at 2:50
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@gekkostate $\exp$ stand for the "exponential" function. One doesn't really know beforehand that $\exp x=e^x$, but it is a very easy thing to see as I'm showing you. The "word" $\sup$ stands for the "supremum" of a set. In the case of real numbers, we define the supremum of a set $A\subseteq \Bbb R$ (if it exists) to be the (unique) number with the following properties: –  Pedro Tamaroff Apr 12 '13 at 2:52
    
$(i)$ For any number $a\in A$, $a\leq x$. That is, $x$ is what we call an upper bound for the elements of $a$. As an example, $2$ is an upper bound for $[0,1)$, because for any $x\in[0,1)$, $x\leq 2$. $(ii)$ If $y$ is any other number with the above property $x\leq y$. That is, $x$ is the least of all the upper bounds. That is why we usually call $\sup$ the "least upper bound" of $A$, and write $\operatorname{lub}(A)$. ${}$ Question: What is $\sup [0,1)$? A tougher one: what is $\sup \{r\in \Bbb Q:r^2<2\}$? –  Pedro Tamaroff Apr 12 '13 at 2:53
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@gekkostate Let $e^h-1=u$. Then $u\to 0\iff e^h-1\to 0$. The limit then states, since $h=\log(u+1)$, $$\lim_{u\to 0}\frac{u}{\log(1+u)}=1$$ which we can prove using the definition of the logarithm I gave you. –  Pedro Tamaroff Apr 13 '13 at 15:25
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$\displaystyle e= \lim_{n \to \infty} \Big(1 + \frac{1}{n} \Big)^{n}$

Let $n = \frac{1}{h}$

Then $ \displaystyle e = \lim_{h \to 0} (1+h)^{\frac{1}{h}}$

which imples that for small $h$, $e^{h} \approx 1+ h$

and $\displaystyle \lim_{h \to 0} \frac{e^{h}-1}{h} = \lim_{h \to 0} \frac{h}{h} = 1$

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HINT :for one definition of e $$e=\lim_{n\rightarrow\infty}(1+\frac1n)^n$$ and use the Binomial expansion

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Use L'Hospital's rule: take a derivative of both the numerator and denominator WRT x and divide the former by the latter inside the limit. You get $e^x \lim_{h\to0}{1\over1}$, since both the numerator and denominator are constant WRT x. You should know what to do from there.

Although I guess that sort of defeats the purpose of doing the limit thing - if we're allowed to use l'Hospital's rule, we usually don't have to prove what the derivative of $e^x$ is either.

To really get into the limit definition, though, in order to to prove that $$\lim_{h\to0} {e^{x+h}-e^{x}\over h} = e^x$$

you will need to prove that $$\forall \varepsilon > 0 \; \exists \; \delta > 0 \ni \left|h\right|<\delta\to\left|{e^{x+h}-e^{x}\over h} - e^x\right|<\varepsilon$$ not just use your sissy limit manipulation tricks.

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Yes, what you're suggesting is circular. –  Pedro Tamaroff Apr 12 '13 at 3:09
    
The problem with your second idea is that there is "no way out" of it. Showing $(e^x)'=e^x$ using limits of difference quotients will boil down to showing $$\frac{e^h-1}{h}\to 1$$ –  Pedro Tamaroff Apr 12 '13 at 3:52
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If you have have a proof for d/dx(ln x), you can always prove d/dx (e^x) by taking d/dx (ln (e^x)) using chain rule. First, we know d/dx (ln(e^x)) = d/dx (x) = 1. Second, using chain rule:

d/dx (ln(e^x)) = (1/e^x)*(d/dx e^x) Using the fact that we know both sides equal 1, then:

(1/e^x)*(d/dx e^x) = 1 Multiplying both sides by e^x gives:

d/dx (e^x) = e^x

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