Symbolic limit of a function of 2 variables, HP49/50

[Gilles Carpentier]

HI,

Here is a program for HP49/50 whch try to find the lim (X,Y)->(a,b) of a function of two variables.
Just put your function on the stack (variables must be X,Y) , and the { a b } limit you search. You will see in return :

lim:number is the limit found by the program

lim: oo (infinite limit)

lim:? no limit found

lim: f(t) no limit, the function returned is the limit in function of t, the  angle of approach.

Must be in RPN and exact mode. Be careful about X Y r t not define as global variables. If you try it, you will see  some limitations but in most cases it works fine.

«

 {'X' 'Y'} DUP ROT ADD = + « SUBST » STREAM

 DUP

 'X=0' SUBST

 'Y=0' SUBST

 IF EVAL DUP '?' SAME THEN

  DROP

  'X=r*COS(t)' SUBST

  'Y=r*SIN(t)' SUBST

  SIMPLIFY

  IFERR 'r=0+0' lim THEN DROP2 '?' END

 ELSE

  NIP

 END

 SIMPLIFY 'Lim' ->TAG

»

'LimXY' STO

For example :

'3*X^2*Y/(X^2+Y^2)'  { 0 0 } LimXY.  Here you get :

lim:0

 '(X^2-Y^2)/(X^2+Y^2)' {0 0}

Lim: '-(2*SIN(t)^2-1)    that means no limit (as the limit depands of the 

angle to approach the point). You can modifiy this if you prefer with '?' 

and no detail

'X*SIN((X+Y)/4)' { 'PI' 'PI' }

Lim:PI

'LOG(X+e^Y)/Sqrt(X^2+Y^2)' { 1 0 }

Lim:LOG(2)

'LN(X+Y)/(X^2+2*X*Y+Y^2-1)' { 1 0 }

Lim:1/2

'(X^2*Y^5)/(X^4+X^2*Y^2+Y^4)' {0 0}

Lim : 0

'8*(X^2*Y^2)/(X^4+Y^4)' {0 0}

Lim: '-((8*SIN(t)^4-8*SIN(t)^2)/(2*SIN(t)^4-2*SIN(t)^2+1))'   @ Mean the 

limit does not exist

SIN(X+Y)/(X+Y)  { 0 0 }

Lim: 1

' (X^2+Y^2-2*X-2*Y)/(X^2+Y^2-2*X+2*Y+2)' { 1 -1 }

Lim: oo   (infinty)

' (X^3+X*Y^2)/(X^2+Y^2)' { 0 0 }

Lim : 0

etc.

Edited: 26 Aug 2012, 10:34 a.m.