Math - Extending L'Hopital's Rule

Extending L'Hopital's Rule for single variable functions to Multivariable Functions.

This method uses the differential form for multivariable functions to find the limit.   

Probably a lot of people have thought of extending L'Hopital's Rule to multivariable functions over the years.
I worked through this intuitive idea in 1980 as an Undergraduate Math major at the University of Minnesota.

Gary R. Lawlor published in 2012 and 2020:
    See: https://arxiv.org/pdf/1209.0363.pdf 
              https://www.tandfonline.com/doi/full/10.1080/00029890.2020.1793635  

Ivlev and Shilin published in 2014:
  See:  https://arxiv.org/pdf/1403.3006.pdf   

Liu and Wang Published in 2010:
  See: https://www.maa.org/sites/default/files/Liu2010.pdf  

Below is a screen shot from  https://arxiv.org/pdf/1403.3006.pdf 

The above seems to suggest that the method proposed has validity. 

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In regards to finding:  lim  f(x1, x2, ... xn)/g(x1, x2, ... xn) as xi--> ci
and f(c1, c2, ... cn) and g(c1, c2, ...cn) = 0  that results in 0/0. 

In the calculations below... in terms of vector calculus. 

Hypothesis-1:

If the gradient of f and gradient of g do not degenerate at the limit point, and if the gradients of f and g are parallel then the limit of f/g, if it exists, is equal to the ratio of the gradients.  The calculations below are expressed in terms of differentials df and dg, but these are also representations of grad f and grad g

Hypothesis-2:

If grad g does not degenerate at the limit point (not equal to zero) but grad f is equal to zero, then we cannot compare grad f and grad g to see if they are parallel. The limit is not known or the limit is indeed zero. 

A favorite and popular quote attributed to theoretical physicist John Archibald Wheeler − “We live on an island surrounded by a sea of ignorance. As our island of knowledge grows, so does the shoreline of our ignorance.” With this in mind consider L’Hopital’s Rule as a small peninsula of knowledge that extends into a multidimensional sea of ignorance. Like a Mandelbrot set with an infinite shoreline, can we extend our boundary of knowledge by generalizing L’Hopital’s Rule to include multivariable functions?  

Proposition:

There are exceptions where this method implies a limit, when in fact the limit does not exist.  See Example 8B below.
However, if you look at the many examples below, they indicate that this method is valid in many cases.  If the limit exists, this method does give the correct limit value K. 

Consider the following analogies (1) and (2). 
These analogies indicate that it is not always possible to make absolute determinations. 

Analogy 1)   y=f(x)     
                            Where: dy/dx = f'(x) = 0
                            1a) could be a minimum
                            1b) could be a maximum 
                            1c) could be an inflection point/a point where the function is horizontal (neither a minimum or a maximum)

Analogy 2) z = f(x, y)
                        Hessian Matrix  H(f(x, y))    =    |   fxx         fxy   |
                                                                                            |   fyx          fyy  |

                      D =   det( H ( f(x, y))) =  fxxfyy - fyzfxy

                      2a) D > 0 and fxx  > 0 then local minimum
                      2b) D > 0 and fxx  < 0 then local maximum 
                      2c) D < 0 then a local saddle point
                      2d) D = 0 then test is inconclusive - point could be local minimum, local maximum or a saddle point                

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This differential form method, like applying L'Hopital's Rule repeatedly, also allows repeated application of the differential operator d(f(x, y, z)) to create multiple levels of the differential form. 
For example:   d(d(f(x, y, z))  yields:  A(x, y, z)dx^2 + B(x, y, z)dxdy + C(x, y, z)dydx + D(x, y, z)dy^2 

This "proof" is not rigorous but provides intuition and motivation for multivariable functions.
The proof does not address exceptions, where this method implies a limit, but it doesn't exist.

Since we are using L'Hopital's Rule here... must have a non-zero derivative in the denominator:
G'(t) is not 0  for all t near to .
This has implications for several of the examples below... but I have not yet worked out the implications.  

Some of the partial derivatives shown above could be zero or tend to zero in the limit.  

Many of these constants could be zero or the partial derivatives could tend to zero in the limit.  

This suggests that if the limit f/g as (x, y, z) -> (a, b, c) exists, then it is K

The ratio of the differential forms cancel each other.  Thus the limit is independent of the path.  

That means we can drop the x''  y'' and z'' terms from the above expression... leaving the ratio below. 

Many of these partial derivatives could be zero or tend to zero in the limit.  

Many of these constants could be zero. 

If the limit exists then the corresponding constants are equal.
Cfxx = Cgxx   and   Cfyx = Cgyx ......  for all constants listed below.
The ratio of the differential forms cancel each other. 
Thus the limit is independent of the path.  

Of course the numerator could be zero, in which case the limit is zero (if it exists).
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To understand how differential terms in the numerator and denominator cancel out, see examples below, or especially see Example 28 on this website titled: "Math - L'Hopital-2"     Example 28 is quite interesting. 
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Here is another motivation/intuition/proof below:  

Probably not mathematically rigorous to multiple by dt, but a physicist would be OK with it!  :) 

If the Limit (df/dg) exists and is K, then the Limit (f/g) is K (if the limit exists).
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When using the "d" operator it is defined as below. 
Where applying the differential operator twice: d(d(f))  =  d^2(f)
Notice: dxdy commutes such that:  dxdy =dydx

Also note when applying the differential operator twice (or multiple times) times:
Adxdy + Bdydx = (A+B)dxdy = (A+B)dydx
This is unlike differential forms where dx^dy = -dy^dx 

See example below for up to 4 repetitions of the differential operator. 

For repeated differentiation of a multivariable function we can define the Zeta Operator as below. 

In the examples below, some of the limits do not exist even tho this method implies they exist. 
In some of these cases this method and Wolfram Alpha agree, and in some cases this method and Wolfram Aalpha disagree.  If the limit does exist, it is likely the value determined by this method.  

For Example 1 above:
Both Wolfram Alpha and Math24.pro  say limit is -1/2 

For Example 2 above:
Both Wolfram Alpha and Math24.pro  say limit is -5

For Example 4 above:
Both Wolfram Alpha and Math24.pro  say limit is  1
3D Math plotter (https://c3d.libretexts.org/CalcPlot3D/index.html) shows a continuous sheet that is 1 at (0, 0). 

For Example 4 we can use the Series expansion for sin(x)  -  sin(y) - see below... 

Example 4 can also be done this way

Example 4 can also be done using the differential in Polar Coordinates 

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Example 5 can also be determined using trigonometric identities - see below...

For Example 6 above:
Both Wolfram Alpha and Math24.pro  say limit is -2
3D Math plotter (https://c3d.libretexts.org/CalcPlot3D/index.html) shows a continuous sheet that is -2 at (0, 0). 

Below is another method computing this limit using the series expansion for cosine.

Another method of computing Example 6 is below.

For Example 7 above: 
3D Math Plotter shows the limit as 2 and there appear to be no rips, holes, tears, or asymptotes (https://c3d.libretexts.org/CalcPlot3D/index.html).  There does not appear to be any wrinkles or dimples.  
Wolfram Alpha Limit Calculator and this Limit Calculator (https://math24.pro/limit2) says the Limit DNE (Does Not Exist). 
Not sure why that would be if the 3D plot appears to be so continuous?

For Example 8 above: 
3D Math Plotter shows the value as -1/2 and there appear to be no rips, holes, tears, or asymptotes (https://c3d.libretexts.org/CalcPlot3D/index.html). 
There is a slight wrinkle or dimple at (x, y, z) = (0, 0, -1/2)
Wolfram Alpha Limit Calculator says the limit is -1/2 - see result below. 

For Example 10 above: 
3D Math Plotter shows the limit generally approaching 0 along several different steep slopes.
But there is a star shaped hole centered on (0,0,0) Even at extreme high resolution there is a star shaped hole.  Possibly, the reason the star shaped hole does not shrink even at high resolution/precision, is that the precision used by the CalcPlot3D might be exceeded, and that results in attempting to divide an extremely small number by another extremely small number thus in effect attempting to divide by zero, and thus cannot be plotted within the area of the star.  
(https://c3d.libretexts.org/CalcPlot3D/index.html)
Wolfram Alpha Limit Calculator and this Limit Calculator Math24.pro (https://math24.pro/limit2) say the Limit DNE (Does Not Exist). 

For Example 11 above: 
3D Math Plotter shows the limit  approaching 0, but with a hole at (0,0,0), where it is undefined (probably due to exceeding its precision). (https://c3d.libretexts.org/CalcPlot3D/index.html)
Wolfram Alpha Limit Calculator and  Math24.pro Limit Calculator (https://math24.pro/limit2) say this limit is 0.

We would think that this limit is 1 because as y->0, this goes to (dx/dx) = 1. 
But look at case 3 below.  We find the limit does not exist. 

For Example 12 above: 
3D Math Plotter shows rips and tears. ((https://c3d.libretexts.org/CalcPlot3D/index.html)
When (x,y) -> (0,0) along certain paths/directions, the limit approaches 1.  
Wolfram Alpha Limit Calculator and this Limit Calculator Math24.pro (https://math24.pro/limit2) says the Limit DNE (Does Not Exist).  

Note:  The df/dg method seems to fail if we let y->0  when calculating 3(y^3)*dy/dx without further analysis. 
I think this is because using L'Hopital's Rule requires the derivative of the denominator to be non-zero in the vicinity of the limit point.
In this case  "dg" is zero along several paths, and therefore L'Hopital's Rule cannot be used.

For Example 13 above: 
3D Math Plotter shows different values depending upon the path taken.  (https://c3d.libretexts.org/CalcPlot3D/index.html)
Along the x-axis (dy/dx=0), it approaches  1.
Along the y-axis (dy/dx = infinity) it approaches -1.   
Wolfram Alpha Limit Calculator and this Limit Calculato Math24.pro  (https://math24.pro/limit2) say the Limit DNE (Does Not Exist).  

For Example 14 above: 
3D Math Plotter shows smooth continuous function that goes to 0 as (x, y)-> (0, 0) (https://c3d.libretexts.org/CalcPlot3D/index.html)
Wolfram Alpha Limit Calculator and this Limit Calculator Math24.pro  (https://math24.pro/limit2) both say the limit is 0. 

Example 14 can also be evaluated using method below.

For Example 15 above: 
3D Math Plotter shows smooth continuous function that goes to 1 as (x, y)-> (0, 0) (https://c3d.libretexts.org/CalcPlot3D/index.html)
Wolfram Alpha Limit Calculator and this Limit Calculator Math24.pro  (https://math24.pro/limit2) both say the limit is 1. 

Example 15 can also be done this way... see below.  

For Example 16 above: 
3D Math Plotter shows smooth continuous function that goes to 1 as (x, y)-> (0, 0) (https://c3d.libretexts.org/CalcPlot3D/index.html)
Wolfram Alpha Limit Calculator and this Limit Calculator Math24.pro  (https://math24.pro/limit2) both say the limit is 1. 

Example 16 can also be done this way... see below. 

For Example 17 above: 
3D Math Plotter shows different values depending upon the path taken.  (https://c3d.libretexts.org/CalcPlot3D/index.html)
Along the x-axis (dy/dx=0), it approaches  0.
Along the y-axis (dy/dx = infinity) it approaches -2.   
Wolfram Alpha Limit Calculator and this Limit Calculato Math24.pro  (https://math24.pro/limit2) say the Limit DNE (Does Not Exist). 

For Example 18 above: 
3D Math Plotter shows smooth continuous function that goes to 1 as (x, y)-> (0, 0) (https://c3d.libretexts.org/CalcPlot3D/index.html)
Wolfram Alpha Limit Calculator and this Limit Calculator Math24.pro  (https://math24.pro/limit2) both say the limit is 1. 

For Example 19 above:
We could say Lim (d^2f/d^2g) --> 2dxdy/(infinity) and therefore the limit does not exist. Maybe that is the right interpretation of the differential forms.
3D Math Plotter shows complicated Vertical sheets with holes (or open tubes) around (x, y) near (0, 0).  (https://c3d.libretexts.org/CalcPlot3D/index.html)
Wolfram Alpha Limit Calculator and this Limit Calculato Math24.pro  (https://math24.pro/limit2) say the Limit DNE (Does Not Exist). 

For Example 20 above: 
3D Math Plotter shows smooth continuous function that goes to  0 as (x, y)-> (0, 0) (https://c3d.libretexts.org/CalcPlot3D/index.html)
Wolfram Alpha Limit Calculator and this Limit Calculator Math24.pro  (https://math24.pro/limit2) both say the limit is  0. 

For Example 21 above: 
3D Math Plotter shows 2 smooth continuous sheets that come together and  "kiss" at point (0, 0, 0). (https://c3d.libretexts.org/CalcPlot3D/index.html)
Wolfram Alpha Limit Calculator says the limit does not exist.  

For Example 22 above: 
*** This example might be the most impressive use of the method for calculating limits using df/dg.   *** 
3D Math Plotter shows a smooth continuous sheet that approaches 1/2 as (x, y) -> (0, 0). (https://c3d.libretexts.org/CalcPlot3D/index.html)
Note: There is a star shaped hole at  (0, 0, 0.5)... but I think that is due to  losing precision as (x, y) -> (0, 0).
Wolfram Alpha Limit Calculator and Limit Calculator Math24.pro  (https://math24.pro/limit2) both say the limit is  1/2. 

For example 22 above, expanding cos(xy) as a series shows a very simple method of calculating the limit.

For Example 23 above: 
3D Math Plotter shows a smooth continuous sheet that goes thru (0, 0, 0). (https://c3d.libretexts.org/CalcPlot3D/index.html)
Wolfram Alpha Limit Calculator and Limit Calculator Math24.pro  (https://math24.pro/limit2) both say the limit is  0. 

For Example 24 above: 
3D Math Plotter shows an oscillating sheet with periodic asymptotes on the x-axis and y-axis.
Near (0, 0, 1) the sheet is smooth.   (https://c3d.libretexts.org/CalcPlot3D/index.html)
Wolfram Alpha Limit Calculator  says the limit is  1. 

For Example 25 above: 
Wolfram Alpha Limit Calculator  says the limit is  0. 

Evaluate Example 26 by expanding as a series: 

Example 26 can also be evaluated by making a substitution and applying L'Hopital's rule for single variable functions - see below. 

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Below: Notes from Math notebook, 1980...