I have found several threads discussing and proving: $ f''(x) = \lim\limits_{h\rightarrow0}\dfrac{f(x+h) - 2 f(x) +f(x-h) }{h^2} $
like: here or there and I am sure, there are probably more discussion pages out there cause this problem is pretty well known.
However I am more interested in the following steps as they are not so often discussed:
$ \lim\limits_{k\to 0} \frac{ \lim\limits_{h\to 0} \frac{ f(x+h) - f(x)}{h} - \lim\limits_{h \to 0} \frac{ f(x+h-k) - f(x-k)}{h} }{k} = \lim\limits_{h\rightarrow0}\dfrac{f(x+h) - 2 f(x) +f(x-h) }{h^2} $
I find very interesting, that how two limits are packed together into one, however I fail finding an easy argument to justify the seeps necessary in between and fill the gaps.
Is there an easy way to do it? What would be a good approach to tackle it? Any constructive comment/answer is appreciated. As always thanks in advance.
