Finding the Limit Symbolically

    The entries in the table show that the coefficients of    are tending to a limit as  .  Thus the "tangent parabola" is  

(5)          .  

The first limit in (5) is well known, it is  

        .

The second limit in (5) is studied in numerical analysis, and is known to be , which can be verified by applying L'hopital's rule using the variable h as follows

         .  

Exploration 1.

Mathematica can find the limit of the difference quotients and obtain the derivatives symbolically.

(c) John H. Mathews 2004