Question

Proof d/dx e^x=e^x
using limit def (1+ 1/h)^h h->infinity

somehow e limit definition implies d/dx e^x=e^x

Answer 1

\[\lim_{h \rightarrow 0} (e ^{x+h} - e ^{x}) / h\]

Answer 2

That is the definition of the derivative of e^x, and you have to proof that that equals e^x

First of all, we have to on it a bit:

\[(e ^{x+h} - e ^{x})/h = e ^{x}(e ^{h}-1)/h\]
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But, then, the first limit is equal to

\[[e ^{x}\lim_{h \rightarrow 0} (e ^{h}-1)/h\]

But, that limit, \[\lim_{h \rightarrow 0} (e ^{h}-1)/h = 1\]

So \[e ^{x}\lim_{h \rightarrow 0} (e ^{h}-1) = e ^{x}.1 = e^x\]

Answer 3

I was able to figure it out ^^
thanks

Answer 4

But we haven´t proof yet that \[\lim_{h \rightarrow 0}(e^h-1) = 1\], so, we have to proof it.
First of all you have to keep in mind the following substitution:
Let t, \[t = 1/(e^h-1) \] , then, \[1/t = e^h-1 \] and, adding 1 at both sides
\[(1/t)+1 = e^h \] and then,taking ln at both sides you get....

Answer 5

you get \[\ln (1/t +1) = h\]

Then, think for a moment about the limit of that if h approaches 0...then ln(1/t +1) has to approach 0....And ln becomes 0 when it argument becomes 1. So, if h approaches to 0, ln has to approach 0 too, and 1/t + 1 has to approach 1, but the only way is if t approaches infinity.

So, if h approaches 0, then t approaches infinity.
Then we are able to proof the limit...

Answer 6

\[\lim_{h \rightarrow o} (e^h - 1)/h =\lim_{t \rightarrow \infty}(1/t)/\ln(1+1/t) \]

So, at this step i have only applied the substitution t that we have written before, and note that at right side we have the limit of t approaching infinity...because the variable change and the consideration about the limits that I did at the last post.

After this point, you have only to work on the expression, and you´ll arrive to the definition of the e number that the exercise suggested

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

At the final step I just used the definition of e number
\[\lim_{t \rightarrow \infty} (1+1/t)^t = e\]

Well, thats all...=)

Answer 7

Oh sorry, I haven´t read you!! Haha, no problem man!
Congratulations then

Answer 8

thanks though haha