Approximation of second derivative of a smooth function

[TheCanadian]

Hi,

I've attached an image of an equation I came across, and the text describes this as an approximation to the second derivative. Everything seems to be exact to me (i.e. not an approximation) if the limit of h was taken to 0. Is that the only reason why it's said to be an approximation or is there any other reason? Also, what exactly is $$ \mathcal{O}(h^2) $$ and why is it included? Isn't the first whole term (i.e. the fraction) the only necessary term to describe the second derivative of $\varphi$?

 

[mathman]

The second derivative is the limit to the fraction as h>0. O(h^2) means the remainder for small h is roughly a constant times h^2.