Math - L'Hopital-2

This is an extension of the ideas on the Math - Extending L'Hopital's Rule Page

Gotta look at that page first.

Example using Zeta Operator *  Ratio of arbitrary polynomials of degree n and m where m < n:

The above seems to take care of a lot of polynomial functions with arbitrary ai and bi. 

Example 27 below shows that there are solutions where the denominator is zero, and therefore the limit does not exist.  

For example 27 above Wolfram Alpha on-line:  lim (x^2+y^2)/(x+y) x->0 y->0   Says this limit is 0.
3D Math Plotter (https://c3d.libretexts.org/CalcPlot3D/index.html) shows two cone shaped sheets that kiss at (0, 0, 0)

Example 28 Below is very interesting. 

For Example 28 Above:
Wolfram Alpha shows this limit as -2:    lim (2sin(xy) - sin(2(x^2+y^2)))/( x^2 + y^2 - sin(xy)) x-> 0 y-> 0 
3D Math Plotter (https://c3d.libretexts.org/CalcPlot3D/index.html) shows a single oscillating sheet
that smoothly crosses z = -2 at (x, y) = (0, 0)