I was reading the paper, "Integration in finite terms" by Maxwell Rosenlicht, and in the last page the author mentions that G.H Hardy has a book on how integration of functions in elementary calculus does comprise a coherent system and that it is not a form of "art" as many people believe.
After reading that I was wondering whether the way limit problems are solved in elementary calculus have any kind of systematic basis. Do we adopt the techniques to solve limit problems ( like multiplying the numerator and denominator by the conjugate) because they "work" or is there some underlying idea that we used to derive these techniques ?
Link to Rosenlicht's paper : http://www.dm.unipi.it/pages/gianni/public_html/Alg-Comp/rosenlicht.pdf