Going back to the previous blog and looking for an alternative second half…
For this to be true
Raise both sides to the th power:
At and it can be seen that both sides equal 1.
Feels like quite a weak proof to me. But I am not in a position to claim expertise. Thoughts, anyone?
One response to “An alternative second half?”
Like your thinking and approach in looking at this from a different perspective (i.e. just using the assumption that there exists a number e such that the derivative of e^x = e^x). Good stuff. Certainly not in a position to claim any expertise at all either but thoughts are that, in general, lim(x->0) f(x).g(x) is only equal to the product of lim(x->0) f(x) and lim(x->0) g(x) if both of these limits are defined already – I think. e.g. (looking at it the other way round) lim(x->0) x/sin(x) = 1 but lim(x->0) x (=0) multiplied by lim(x->0) 1/sin(x) (undefined) is clearly not 1 (I’m not sure whether, technically, it’s 0 or undefined but anyway – it’s certainly not 1). So, even if lim(d,n->0) ((1+n)^(d/n)-1)/d is defined (and equal to 1) I’m not sure it necessarily follows that lim(n->0) of (1+n)^(x/n) is defined and therefore equal to e^x.