Twice differentiable

Asked Mar 01 '17 at 10:46

Active Mar 01 '17 at 11:14

Viewed 668 times

While reading in Wikipedia (Derivative # Higher derivatives) I encountered this definition:

I don't understand why $f(x+h)$ equals to the expression in the right side (where is this come from?).
Plus, how did they get to the limit expression? Have they moved the right side to the left and divide by $h^2$? I'm having trouble seeing the logic behind it.
In addition, can I turn the last expression within the limit to an expression without $f'(x)$ and $f''(x)$ - I mean, to remain only with $f(x)$?
Lastly, if a function has the limit above does it mean it is twice differentiable?

Thanks in advance!

edited Mar 01 '17 at 11:14

CiaPan

asked Mar 01 '17 at 10:46

user401575

1

This is Taylor expansion built at $h=0$. – Claude Leibovici Mar 01 '17 at 11:09
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That is not a definition.but rather an implication of Taylor's series for the function $f$ at $x$. – CiaPan Mar 01 '17 at 11:16
See http://math.stackexchange.com/q/1809293/72031 – Paramanand Singh Mar 01 '17 at 16:44

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