The mistake is at the beginning:
The following is a part of a current I(s) (in -space):
What am I doing wrong in using this formula for finding and :
; where is a pole.
I easily find A, but not B:
end of PROBLEM
The solution (which is correct) is
What am I doing wrong inside the PROBLEM section? Or is the formula for (or ) wrong?
Yes, I wrote that. For some reason Math Help Forum wasn't compiling Latex at the time I wrote it. I'm still having trouble with their Latex and so I won't use it.
No, the formula for A is correct. You divide B by L because sL is in the denominator of F(s).
You don't multiply by L; B/L gives the correct answer.
Why are you bothering with the formalism of limits? This is just a (very trivial) partial fraction decomposition you should have learned in precalculus --
You have the following after clearing fractions:
(1). A * (R + sL) + B * (s) = U
Let s = 0 => A = U/R
Let s = -R/L => B = -UL/R
Therefore, after substitution for A and B into your original decomposition, we obtain:
(A/s) + (B/(R + sL)) = (U/(Rs)) - (UL/(R(R+sL))).
The answer is easily verifiable by recombining fractions.
By the way, this is known as the Heaviside "cover up" method. Here is a helpful resource you should consider: http://www.math-cs.gordon.edu/course.../heavyside.pdf. SO now, you can forget about this limit formalism and just "cover up" the factors (only applies when all factors are linear).