hello, I need the proof to show that:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

e^x = \lim_{n\rightarrow\infty} (1+x/n)^n

[/tex]

Here's what I was able to come up with so far:

since the derivative of [tex]e^x[/tex] is also [tex]e^x[/tex],

then let f(x) = [tex]e^x[/tex]

thus:

D(f(x)) = [tex]\lim_{n\rightarrow\infty} \frac{f(x+h) - f(x)}{h} = \lim_{n\rightarrow\infty} \frac{e^{x+h} - e^x}{h} = e^x\lim_{n\rightarrow\infty} \frac{e^h - 1}{h}[/tex]

so for the derivative of [tex]e^x[/tex] to equal itself,

[tex]\lim_{n\rightarrow\infty} \frac{e^h - 1}{h} = 1[/tex]

so for small values of h, we can write:

[tex]e^h - 1 = h[/tex]

and so

[tex] e = (1+h)^{1/h}[/tex]

Replacing h by 1/n, we get:

[tex] e = (1 + 1/n)^n[/tex]

As n gets larger and approaches infinity, we get:

[tex]e = \lim_{n\rightarrow\infty} (1+1/n)^n[/tex]

so, how do I get:

[tex]

e^x = \lim_{n\rightarrow\infty} (1+x/n)^n

[/tex]

Also, is it true that:

[tex](1+x/n)^n \leq e^x[/tex] and

[tex](1-x/n)^n \leq e^{-x}[/tex] for every natural n and every x element of X?

How can I prove this?

thanks!

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# Proof for exponentials

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