Is it okay to perform operations under the limit sign just as we would if the limit sign were not there? I'm really asking this to better understand how to derive the value of e.(adsbygoogle = window.adsbygoogle || []).push({});

e is defined so that its exponential function is its own derivative. I won't go through all of it (since I'm sure most of the people on this site are familiar) but that amounts to saying that

lim ((e^h-1)/h) = 1

h->0

Now, if we simply had ((e^h-1)/h) = 1 without the limit and wanted to solve for e, we would get

e = (h+1)^(1/h)

Indeed, the true value of e, putting the limit back in, is

e = lim (h+1)^(1/h)

h->0

I have tried this out with a few other functions and it always turns out that performing operations under the limit sign as if it weren't even there and then slapping it back on at the end gives the correct answer, but I just can't convince myself that it's "legal". Can anyone help convince me?

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# Solving under the limit sign

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