second derivative as simple limit

If the real function f is twice differentiableMathworldPlanetmathPlanetmath in a neighbourhood of  x=x0,  then

f′′(x0)=limh0f(x0+2h)-2f(x0+h)+f(x0)h2. (1)

Proof.  The right hand side of the asserted equation is of the indeterminate form 00.  Using’Hôpital’s rule, we obtain

limh0f(x0+2h)-2f(x0+h)+f(x0)h2 =limh0f(x0+2h)2-2f(x0+h)2h-f(x0)h+f(x0)h
= 2lim2h0f(x0+2h)-f(x0)2h-limh0f(x0+h)-f(x0)h
= 2f′′(x0)-f′′(x0)
Title second derivative as simple limit
Canonical name SecondDerivativeAsSimpleLimit
Date of creation 2013-03-22 19:00:00
Last modified on 2013-03-22 19:00:00
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Result
Classification msc 26A24
Synonym second derivative as limit
Related topic DifferenceQuotient
Related topic ImproperLimits