L'hospitals Rule with severable variables

[Daniel]

I need to know how L'hospitals Rule applies to functions of several variables.

For example

Lim(x,y)->(0,0) (2x^2-y^2)/(2x^2+y^2)

if i apply L'Hospitals Rule to that does it simply equal
(4x-2y)/(4x+2y)?

or is there something that has to be done involving partials; OR does it juts not apply to functions of several variables.

[David Moran]

"Daniel" <drd...@vt.edu> wrote in message
news:23218138.1134502986...@nitrogen.mathforum.org...

There is not a version of L'Hospital's rule for multivariate functions. The
way to do these is approach the origin from various lines (i.e. y=x, y=2x,
y=3x, ....). If these are all the same, the limit exists, otherwise it does
not. I don't think the limit above exists, but I've not looked at it in
detail.

Dave

[Arturo Magidin]

In article <2JFnf.28828$QW2.16957@dukeread08>,

David Moran <dmor...@cox.net> wrote:
>
>"Daniel" <drd...@vt.edu> wrote in message
>news:23218138.1134502986...@nitrogen.mathforum.org...
>>I need to know how L'hospitals Rule applies to functions of several
>>variables.
>>
>> For example
>>
>> Lim(x,y)->(0,0) (2x^2-y^2)/(2x^2+y^2)
>>
>> if i apply L'Hospitals Rule to that does it simply equal
>> (4x-2y)/(4x+2y)?
>>
>> or is there something that has to be done involving partials; OR does it
>> juts not apply to functions of several variables.
>
>There is not a version of L'Hospital's rule for multivariate functions. The
>way to do these is approach the origin from various lines (i.e. y=x, y=2x,
>y=3x, ....). If these are all the same, the limit exists,

The last assertion does not follow. It is possible for the limits to
exist along all lines, be equal, and yet for the limit not to exist.
You need the limit along ALL possible approaches to the origin to
exist and be equal.

> otherwise it does
>not. I don't think the limit above exists, but I've not looked at it in
>detail.

It does not; along the line y=x we obtain the limit of x^2/3x^2, which
is equal to 1/3. Along the line y=0, however, we get the limit of
2x^2/2x^2 which is equal to 1.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
mag...@math.berkeley.edu

[Arturo Magidin]

In article <dnn9qe$143l$1...@agate.berkeley.edu>,

Arturo Magidin <mag...@math.berkeley.edu> wrote:
>In article <2JFnf.28828$QW2.16957@dukeread08>,
>David Moran <dmor...@cox.net> wrote:

Sorry, I meant to give an example:

>>The way to do these is approach the origin from various lines (i.e. y=x, y=2x,
>>y=3x, ....). If these are all the same, the limit exists,
>
>The last assertion does not follow. It is possible for the limits to
>exist along all lines, be equal, and yet for the limit not to exist.
>You need the limit along ALL possible approaches to the origin to
>exist and be equal.

Consider f(x,y) = (xy^2)/(x^2+y^4), and the limit as (x,y)->(0,0).

If we approach along y=mx, we have
f(x,mx) = (m^2x^3)/(x^2+m^4x^4) = (m^2x)/(1+m^4x^2)
which goes to 0 as x->0. If x=0, then the function f(0,y) is 0 at all
points other than (0,0), so the limit is equal to 0. So the limit
along any line is equal to 0.

However, the limit does not exist: if we approach (0,0) along the
parabola x=y^2, then we have

f(y^2,y) = y^4/y^4+y^4 = y^4/2y^4, which has a limit of 1/2 as y->0.

[David Moran]

"Arturo Magidin" <mag...@math.berkeley.edu> wrote in message
news:dnna1k$148t$1...@agate.berkeley.edu...

Sorry, just bad wording on my part

[Dave L. Renfro]

David Moran wrote:

> There is not a version of L'Hospital's rule for multivariate
> functions. The way to do these is approach the origin from
> various lines (i.e. y=x, y=2x, y=3x, ....). If these are all
> the same, the limit exists, otherwise it does not. I don't
> think the limit above exists, but I've not looked at it in
> detail.

This seems to be a commonly held belief (see posts below) which
didn't hold up when I did some digging into the literature (see
the references further below).

Advanced Calculus Discussion Forum: "L'Hopital's rule"
http://tinyurl.com/bd3b7
http://tinyurl.com/ar9o7

sci.math thead: "L'Hopital's Rule in 2-D ?"
7 posts (November 9-16, 1994)
http://groups.google.com/group/sci.math/msg/5a0b7390ab011e32

sci.math thread: "L'Hopital and Multivariable Calculus"
5 posts (November 11-17, 1998)
http://groups.google.com/group/sci.math/msg/fd0a1f36af2f2f92

sci.math thread: "L'Hopital for R^n"
12 posts (January 4-11, 2003)
http://groups.google.com/group/sci.math/msg/8926aa9135e4ea07

[1] Eugen Dobrescu and Ioan Siclovan, "Considerations on
functions of two variables" (Romanian), Analele Universitatii
Timisoara Seria Stiinte Matematica-Fizica 3 (1965), 109-121.
[MR 34 #5998; Zbl 166.31502]
http://www.emis.de/cgi-bin/Zarchive?an=0166.31502

[2] A. I. Fine and S. Kass, "Indeterminate forms for multi-place
functions", Annales Polonici Mathematici 18 (1966), 59-64.
[MR 32 #7680; Zbl 137.03603]
http://www.emis.de/cgi-bin/Zarchive?an=0137.03603
http://journals.impan.gov.pl/cgi-bin/shvold?ap18

[3] Ira Rosenholtz, "A topological mean value theorem for the
plane", American Mathematical Monthly 98 (1991), 149-154.
[MR 91m:26014; Zbl 741.26003]
http://www.emis.de/cgi-bin/MATH-item?0741.26003

[4] Tadeusz Wazewski, "Une généralisation des théoremes sur les
accroissements finis au cas des espaces abstraits.
Applications", Bull. Int. Acad. Polon. Sci. Lett.,
Cl. Sci. Math. Natur., Ser. A 1949, 183-185.
[MR 12,508a; Zbl 41.23301
http://www.emis.de/cgi-bin/Zarchive?an=0041.23301

[5] Tadeusz Wazewski, "Une généralisation des théorèmes sur les
accroissements finis au cas des espaces de Banach et
application à la généralisation du théorème de l'Hôpital"
Ann. Soc. Polon. de Math. 24 (1951), 132-147.
[MR 15,717g; Zbl 52.11302]
http://www.emis.de/cgi-bin/Zarchive?an=0052.11302

[6] Tadeusz Wazewski, "Une modification due théorème de l'Hôpital
liée au problème du prolongement des intégrales des équations
différentielles", Annales Polonici Mathematici 1 (1954),
1-12. [MR 16,118e; Zbl 56.11402
http://www.emis.de/cgi-bin/Zarchive?an=0056.11402
http://matwbn.icm.edu.pl/ksiazki/apm/apm01/apm0101.pdf

[7] William H. Young, "On indeterminate forms", Proceedings of
the London Mathematical Society (2) 8 (1910), 40-76.
[JFM 40.0334.01]
http://www.emis.de/cgi-bin/JFM-item?40.0334.01

(from p. 71) "We now pass to one or two generalisations to
more than one variable. It has commonly, but erroneously,
been supposed, that such generalisations did not exist."

Dave L. Renfro