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If f is a function that is two times differentiable at x = a then:

$\lim\limits_{h \to 0} \frac{f(a+h)-f(a)-hf'(a)}{h^2/2}=f''(a)$

I don't know how to prove or disprove this. I know I have to use the definition of derivation but I have no clue how to go on. Sorry for my grammar. English it's not my native language.

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It is given that $f''(a)$ exists which means that $f'(x)$ exists in a neighborhood of $a$. Thus we have by an application of L'Hospital's Rule $$\lim_{h \to 0}\frac{f(a + h) - f(a) - hf'(a)}{h^{2}/2} = \lim_{h \to 0}\frac{f'(a + h) - f'(a)}{h} = f''(a)$$

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Hint: Use L'hospital rule twice.

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  • $\begingroup$ I am not going to downvote, but the answer is wrong. We need to use L'Hospital only once. The conditions on $f$ don't allow us to use L'Hospital twice. $\endgroup$ – Paramanand Singh Aug 24 '14 at 9:02
  • $\begingroup$ I have already given an answer. I just wanted to say that your hint should be modified to "Use L'Hospital rule". The use of word "twice" is technically wrong. Anyway I don't want to too nit-picky about this and hence I have not given up/down vote. I have seen many people using LHR multiple times in such problems and it is better to let them know when the rule can or can not be applied. $\endgroup$ – Paramanand Singh Aug 25 '14 at 10:25
  • $\begingroup$ Thank you for the explanation, my friend. $\endgroup$ – DeepSea Aug 25 '14 at 11:21

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