Eliminating radicals from denominator or numerator of an expression

Question

I need to find the limit when r2 tends to r1 of the following expression:

(K2 r1^2 + K1 r1 r2 - 
   Sqrt[r1 r2 (K1^2 r1 r2 + K2^2 r1 r2 + K1 K2 (r1^2 + r2^2))]) /
(-K2 r1 r2 - K1 r2^2 + 
   Sqrt[r1 r2 (K1^2 r1 r2 + K2^2 r1 r2 + K1 K2 (r1^2 + r2^2))])

For this reason I want to rationalize the numerator or denominator to see if this helps me find the limit. I am interested in using this procces because when r1 set equal to r2, the expression takes the indeterminate form of zero/zero.

Is there any method to rationalize the expression?

Also I have tried to calculate the limit, but Mathematica reports that the limit is 1. But numerically the result is another one. This different result is shown in the code and in the image. That is, evaluating

(K2 r1^2 + K1 r1 r2 - 
   Sqrt[r1 r2 (K1^2 r1 r2 + K2^2 r1 r2 + K1 K2 (r1^2 + r2^2))]) / 
(-K2 r1 r2 - K1 r2^2 + 
   Sqrt[r1 r2 (K1^2 r1 r2 + K2^2 r1 r2 + K1 K2 (r1^2 + r2^2))]) /. 
{K1 -> 20, K2 -> 30, r1 -> 0.1, r2 -> 0.10001}

gives

Answer 1

Limit will need assumptions on the parameters in this case.

expr = (K2 r1^2 + K1 r1 r2 - 
    Sqrt[r1 r2 (K1^2 r1 r2 + K2^2 r1 r2 + 
        K1 K2 (r1^2 + r2^2))])/(-K2 r1 r2 - K1 r2^2 + 
    Sqrt[r1 r2 (K1^2 r1 r2 + K2^2 r1 r2 + K1 K2 (r1^2 + r2^2))]);

Limit[expr, r1 -> r2, Assumptions -> {K1 > 0, K2 > 0, r2 > 0}]

(* Out[51]= K2/K1 *)

I believe that what goes astray, in presence of parameters, is a failure to correctly assess branch cut related issues. I'm not entirely positive of that however.

Answer 2

The expression evaluates to -1 for r1 == r2 or in the limit. Have you previously assigned values to K1 and/or K2?

expr = (K2 r1^2 + K1 r1 r2 - 
     Sqrt[r1 r2 (K1^2 r1 r2 + K2^2 r1 r2 + K1 K2 (r1^2 + r2^2))])/(-K2 r1 r2 -
      K1 r2^2 + Sqrt[r1 r2 (K1^2 r1 r2 + K2^2 r1 r2 + K1 K2 (r1^2 + r2^2))]);

expr /. r1 -> r2 // Simplify

(* -1 *)

expr /. r2 -> r1 // Simplify

(* -1 *)

Limit[expr, r1 -> r2]

(* -1 *)

Limit[expr, r2 -> r1]

(* -1 *)

Answer 3

If you replace the Sqrt with Surd (which gives the positive value of the square root), then the formula simplifies:

FullSimplify[(K2 r1^2 + K1 r1 r2 - 
    Surd[r1 r2 (K1^2 r1 r2 + K2^2 r1 r2 + K1 K2 (r1^2 + r2^2)), 
     2])/(-K2 r1 r2 - K1 r2^2 + 
    Surd[r1 r2 (K1^2 r1 r2 + K2^2 r1 r2 + K1 K2 (r1^2 + r2^2)), 2])]

The answer is:

So you can see that the limit as r1->r2 makes the numerator of the fraction zero, leaving only the -1 term.