Proving $\displaystyle\lim_{h\to0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}=f''(x)$

real-analysis calculus

We write the Taylor series:

$$f(x+h)=f(x)+hf'(x)+\frac{h^2}{2}f''(x)+h^2\epsilon(h)\tag{1}$$ and $$f(x-h)=f(x)-hf'(x)+\frac{h^2}{2}f''(x)+h^2\epsilon'(h)\tag{2}$$ where $$\lim_{h\to0}\epsilon (h)=\lim_{h\to0}\epsilon' (h)=0$$ so add the two equalities $(1)$ and $(2)$ and you find the result easily.

Related

Divisor function asymptotics
Limit $\lim_{n \to \infty} \frac {a_n}{2^{n-1}}=\frac 4{\pi}$ for $a_{n+1}=a_n+\sqrt{1+a_n^2}$ and $a_0=0$
Show that orthogonal matrices are diagonalizable
What's the quickest way to see that the subset of a set of measure zero has measure zero?
product of harmonic functions
Volterra integral equation of second type solve using resolvent kernel
the 2-D divergence theorem and Green's Theorem
Use induction to prove that $ 1 + \frac {1}{\sqrt{2}} + \frac {1}{\sqrt{3}} .... + \frac {1}{\sqrt{n}} < 2\sqrt{n}$
Limit points of $\cos n$.
Is the set of all bounded sequences complete?
How to define own group action in GAP?
If $A$ and $B$ are positive-definite matrices, is $AB$ positive-definite?