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[parent] second derivative as simple limit (Result)

If the real function $f$ is twice differentiable in a neighbourhood of $x = x_0$ , then

$\displaystyle f''(x_0) \;=\; \lim_{h \to 0}\frac{f(x_0\!+\!2h)-2f(x_0\!+\!h)+f(x_0)}{h^2}.$ (1)

Proof. The right hand side of the asserted equation is of the indeterminate form $\frac{0}{0}$ . Using l'Hôpital's rule, we obtain

$\displaystyle \lim_{h \to 0}\frac{f(x_0\!+\!2h)-2f(x_0\!+\!h)+f(x_0)}{h^2}$ $\displaystyle \;=\; \lim_{h \to 0}\frac{f'(x_0\!+\!2h)\cdot2-2f'(x_0\!+\!h)}{2h}-\frac{f'(x_0)}{h}+\frac{f'(x_0)}{h}$    
  $\displaystyle \;=\;2\lim_{2h \to 0}\frac{f'(x_0\!+\!2h)-f'(x_0)}{2h}-\lim_{h \to 0}\frac{f'(x_0\!+\!h)-f'(x_0)}{h}$    
  $\displaystyle \;=\; 2f''(x_0)-f''(x_0)$    
  $\displaystyle \;=\; f''(x_0).$    



"second derivative as simple limit" is owned by pahio.
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See Also: difference quotient, improper limits

Other names:  second derivative as limit
Keywords:  second derivative

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Cross-references: indeterminate form, equation, right hand side, proof, neighbourhood, twice differentiable, real function

This is version 6 of second derivative as simple limit, born on 2009-08-19, modified 2009-08-22.
Object id is 11868, canonical name is SecondDerivativeAsSimpleLimit.
Accessed 382 times total.

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AMS MSC: 26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)
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