second derivative as simple limit

If the real function $f$ is twice differentiable in a neighbourhood of  $x=x_{0}$,  then

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

Proof.  The right hand side of the asserted equation is of the indeterminate form $\frac{0}{0}$.  Using http://planetmath.org/node/2657l’Hôpital’s rule, we obtain

 $\displaystyle\lim_{h\to 0}\frac{f(x_{0}\!+\!2h)-2f(x_{0}\!+\!h)+f(x_{0})}{h^{2}}$ $\displaystyle\;=\;\lim_{h\to 0}\frac{f^{\prime}(x_{0}\!+\!2h)\cdot 2-2f^{% \prime}(x_{0}\!+\!h)}{2h}-\frac{f^{\prime}(x_{0})}{h}+\frac{f^{\prime}(x_{0})}% {h}$ $\displaystyle\;=\;2\lim_{2h\to 0}\frac{f^{\prime}(x_{0}\!+\!2h)-f^{\prime}(x_{% 0})}{2h}-\lim_{h\to 0}\frac{f^{\prime}(x_{0}\!+\!h)-f^{\prime}(x_{0})}{h}$ $\displaystyle\;=\;2f^{\prime\prime}(x_{0})-f^{\prime\prime}(x_{0})$ $\displaystyle\;=\;f^{\prime\prime}(x_{0}).$
Title second derivative as simple limit SecondDerivativeAsSimpleLimit 2013-03-22 19:00:00 2013-03-22 19:00:00 pahio (2872) pahio (2872) 9 pahio (2872) Result msc 26A24 second derivative as limit DifferenceQuotient ImproperLimits