L'Hospital's rule for functions several variables

On Aug 11, 4:29 am, novis wrote:
dear all, is there any concept of L'Hospital's rule for functions
several variables? If yes, then what is the corresponding analogue for
function of several variables?
Thanks

For example function of two independent variables x, y,
this rule would go like the following:

f(x_0,y_0)=0, g(x_0,y_0)=0 , lim_[(x,y)-(x_0,y_0)] f(x,y)/ g(x,y) = ?

lim_[(x,y)-(x_0,y_0)] f(x,y)/ g(x,y) =

lim_[(x,y)-(x_0,y_0)] (f_x+f_y (dy/dx))/(g_x+g_y (dy/dx))=

lim_[(x,y)-(x_0,y_0)] (f_x (dx/dy)+f_y)/(g_x (dx/dy)+g_y)=

In order limit to exist these must be independent of dy/dx or dx/dy
(limit must be independent of the path how to approach
the point (x_0,y_0)). Condition f_x / g_x = f_y / g_y must thus
be fullfilled in order that unique limit exist.

lim_[(x,y)-(x_0,y_0)] f(x,y)/ g(x,y) =

lim_[(x,y)-(x_0,y_0)] f_x / g_x =

lim_[(x,y)-(x_0,y_0)] f_y / g_y.

I thing that this can be similarly be generalized to functions of
several independent variables, but now the condition of existence
of the unique limit would be little different.


Hannu

On Aug 11, 12:12 pm, Hannu Poropudas wrote:
On Aug 11, 4:29 am, novis wrote:

dear all, is there any concept of L'Hospital's rule for functions
several variables? If yes, then what is the corresponding analogue for
function of several variables?
Thanks

For example function of two independent variables x, y,
this rule would go like the following:

f(x_0,y_0)=0, g(x_0,y_0)=0 , lim_[(x,y)-(x_0,y_0)] f(x,y)/ g(x,y) = ?

lim_[(x,y)-(x_0,y_0)] f(x,y)/ g(x,y) =

lim_[(x,y)-(x_0,y_0)] (f_x+f_y (dy/dx))/(g_x+g_y (dy/dx))=

lim_[(x,y)-(x_0,y_0)] (f_x (dx/dy)+f_y)/(g_x (dx/dy)+g_y)=

In order limit to exist these must be independent of dy/dx or dx/dy
(limit must be independent of the path how to approach
the point (x_0,y_0)). Condition f_x / g_x = f_y / g_y must thus
be fullfilled in order that unique limit exist.

lim_[(x,y)-(x_0,y_0)] f(x,y)/ g(x,y) =

lim_[(x,y)-(x_0,y_0)] f_x / g_x =

lim_[(x,y)-(x_0,y_0)] f_y / g_y.

I thing that this can be similarly be generalized to functions of
several independent variables, but now the condition of existence
of the unique limit would be little different.

Hannu

For example, does (x^2 - y^2)/(x^2 + y^2) have a unique value at
origin? If y/x = m as a given direction, then it may be (1 - m^2)/(1 +
m^2).

In article om,
Hannu Poropudas wrote:

On Aug 11, 4:29 am, novis wrote:
dear all, is there any concept of L'Hospital's rule for functions
several variables? If yes, then what is the corresponding analogue for
function of several variables?
Thanks

For example function of two independent variables x, y,
this rule would go like the following:

f(x_0,y_0)=0, g(x_0,y_0)=0 , lim_[(x,y)-(x_0,y_0)] f(x,y)/ g(x,y) = ?

lim_[(x,y)-(x_0,y_0)] f(x,y)/ g(x,y) =

lim_[(x,y)-(x_0,y_0)] (f_x+f_y (dy/dx))/(g_x+g_y (dy/dx))=

lim_[(x,y)-(x_0,y_0)] (f_x (dx/dy)+f_y)/(g_x (dx/dy)+g_y)=

In order limit to exist these must be independent of dy/dx or dx/dy
(limit must be independent of the path how to approach
the point (x_0,y_0)). Condition f_x / g_x = f_y / g_y must thus
be fullfilled in order that unique limit exist.

lim_[(x,y)-(x_0,y_0)] f(x,y)/ g(x,y) =

lim_[(x,y)-(x_0,y_0)] f_x / g_x =

lim_[(x,y)-(x_0,y_0)] f_y / g_y.

I thing that this can be similarly be generalized to functions of
several independent variables, but now the condition of existence
of the unique limit would be little different.


Hannu

It's easy to see lim_[(x,y)-(x_0,y_0)] f_x / g_x =
lim_[(x,y)-(x_0,y_0)] f_y / g_y = L does not imply
lim_[(x,y)-(x_0,y_0)] f(x,y)/ g(x,y) = L, or even that the last limit
exists.

On Aug 11, 9:12 am, Hannu Poropudas wrote:
On Aug 11, 4:29 am, novis wrote:

dear all, is there any concept of L'Hospital's rule for functions
several variables? If yes, then what is the corresponding analogue for
function of several variables?
Thanks

For example function of two independent variables x, y,
this rule would go like the following:

f(x_0,y_0)=0, g(x_0,y_0)=0 , lim_[(x,y)-(x_0,y_0)] f(x,y)/ g(x,y) = ?

lim_[(x,y)-(x_0,y_0)] f(x,y)/ g(x,y) =

lim_[(x,y)-(x_0,y_0)] (f_x+f_y (dy/dx))/(g_x+g_y (dy/dx))=

lim_[(x,y)-(x_0,y_0)] (f_x (dx/dy)+f_y)/(g_x (dx/dy)+g_y)=

In order limit to exist these must be independent of dy/dx or dx/dy
(limit must be independent of the path how to approach
the point (x_0,y_0)). Condition f_x / g_x = f_y / g_y must thus
be fullfilled in order that unique limit exist.

lim_[(x,y)-(x_0,y_0)] f(x,y)/ g(x,y) =

lim_[(x,y)-(x_0,y_0)] f_x / g_x =

lim_[(x,y)-(x_0,y_0)] f_y / g_y.

I thing that this can be similarly be generalized to functions of
several independent variables, but now the condition of existence
of the unique limit would be little different.

Hannu

is hopital not hospital you moron

On Aug 12, 1:55 pm, gimp wrote:
On Aug 11, 9:12 am, Hannu Poropudas wrote:

On Aug 11, 4:29 am, novis wrote:

dear all, is there any concept of L'Hospital's rule for functions
several variables? If yes, then what is the corresponding analogue for
function of several variables?
Thanks
----
Hannu

is hopital not hospital you moron

If the French circonflex was not emphasized here im maths, is it so
bad?

Narasimham wrote:

On Aug 12, 1:55 pm, gimp wrote:


On Aug 11, 9:12 am, Hannu Poropudas wrote:



On Aug 11, 4:29 am, novis wrote:


dear all, is there any concept of L'Hospital's rule for functions
several variables? If yes, then what is the corresponding analogue for
function of several variables?
Thanks


----


Hannu


is hopital not hospital...



If the French circonflex was not emphasized here im maths, is it so
bad?


Many different spellings (capitalized ell or not, circumflex or not,
ess or not) are found in the literature and, I suspect, during the
man's lifetime. The only wrong version would have both a circumflex and
an ess.

--
Stephen J. Herschkorn
Math Tutor on the Internet and in Central New Jersey and Manhattan