Math-L'Hopital-4

Example 46

For this example, we use online tools SymPy and/or Wolfram Alpha to calculate and evaluate differentials. 

Below shows SymPy or 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. 

Now compute the limit of the second order differentials for f(x, y, z) 

Now compute the limit of the second order differentials for g(x, y, z) 

This results in:  

Evaluate the limit using Wolfram Alpha:

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Example 47: 

Define in SymPy function f_x_y
Compute its first derivative
Then compute its 5th order derivatives at limit point (x, y) --> (0, 0)

Define in SymPy function g_x_y
Compute its first derivative
Then compute its 5th order derivatives at limit point (x, y) --> (0, 0)

Now evaluate the 5th order derivatives of f(x, y) and g(x, y) at the limit point. 

Use SymPy to compute the limit of the ratio of f(x, y)/g(x, y) as (x, y) --> (0, 0).