Math-L'Hopital-3

For Example 32 above:
Wolfram Alpha says lim (x^2 + y^2 - sin(x^2 + y^2))/(x^2 + y^2) ^ 3  x-> 0 y-> 0    is   1/6

Solving Example 32 by expanding sin(t) into a series below.... 

Solving Example 32 by letting u = x^2 + y^2  

For Example 33 above:
Wolfram Alpha says lim (2x^2 - xy - y^2)/( x^2 - y^2) as x-> 1 y-> 1     is     3/2   

Or we can do example 33 like this - below.  

For Example 34 above:
WolframAlpha says this limit:  lim (x^2 + y^2)/ (   (x^2 + y^2 + 1)^(1/2) -1)   x-> 0    y-> 0     is    2  

Or we can do Example 35 this way.  

Or we can do Example 36 by converting to polar coordinates and letting r-> 0  (see below) 

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Example 40B below shows a 'standard' way of evaluating the limit in Example 40

Example 41B below shows evaluating example 41 by substituting u = (......)

WolframALpha says:
Lim (  2- 2cos(2x) +  y^2 + cos(2x)y^2 ) / (  12x^2 + x^2cos(2x)  + 12  + cos(2x) - 12cos(y)  -  cos(2x)cos(y)  ) (x, y) -> (0,0)   =  4/13

Example 44: Using SymPy to calculate 4th order differentials for f and g.   See this link:  https://live.sympy.org 

Where expr and expr2 is defined in SymPy as:

Fourth order partial derivatives of f(x, y) evaluated at (x, y) = (0,0) as calculated by SymPy: 

Fourth order partial derivatives of g(x, y) evaluated at (x, y) = (0,0) as calculated by SymPy: 

This results in ...
(Note there are 6 dx^2dy^2 terms, 4 dx^3dy terms and 4 dxdy^3 terms)

Using SymPy to calculate the limit: 

Using Wolfram-Alpha to calculate the limit:  

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Example 45:

For this example, we use online tool Wolfram Alpha to calculate and evaluate differentials.

Now calculate the mixed second order partial derivatives. 

The above shows Wolfram Alpha results for: fx, fy, fz, fxx, fyy, fzz, fxy, fxz, and fyz. Not shown are: gx, gy, gz, gxx, gyy, gzz, gxy, gxz, and gyz.

The limit of each of the first partial derivatives of f(x, y, z) and g(x, y, z) as (x, y, z) -> (0,0,0) is zero. 

Below are results from Wolfram Alpha for Lim of each of the second partial derivatives of f(x, y, z) and g(x, y, z) as (z, y, z) -> (0,0,0)

Plugging in the values for each of these limits results in: 

Below uses Wolfram Alpha to evaluate the limit of f(x, y, z)/g(x, y, z).