The problem involves evaluating the limit of the function R(x)/x^2 as x approaches infinity, where R(x) is defined as the integral from 0 to x of the function sqrt(1+t^2). Participants are tasked with finding this limit analytically rather than through numerical evaluation.
The discussion is ongoing, with various methods being explored, including L'Hospital's rule and alternative approaches to evaluate the limit. Participants are questioning assumptions and clarifying steps, indicating a productive exchange of ideas without reaching a consensus.
There is a focus on the behavior of the function as x approaches infinity, with participants expressing uncertainty about the best method to apply and the implications of their calculations. The conversation reflects a mix of approaches and interpretations regarding the limit and the use of L'Hospital's rule.
LCKurtz said:Use L'Hospital's rule remembering how to differentiate an integral as a function of its upper limit.
armolinasf said:Let me know if this works: applying L'H I get x/(1+x^2) * 1/2 if I set x equal to infinity wouldn't that give me inf/inf=1*1/2 which would give me the correct limit?
armolinasf said:you're right, my mistake.
taking derivatives I get sqrt(1+x^2)/2x. So, can't I factor out 1/2 leaving me with sqrt(1+x^2)/x * 1/2. If I take the limit it should be infinity/infinity times 1/2 which is the same 1*1/2?
armolinasf said:sqrt(1+x^2)/x is indeterminate when x=inf. So if I were apply l'Hopital again I get into this repeating pattern where I wind up at the same thing. For example, applying L'H to sqrt(1+x^2)/x gives x sqrt(1+x^2)/x and I'm back where I started.
By the way, the actual equation is sqrt(1+x^2)/2x I'm wondering if it's possible to factor out a 1/2 and then apply L'H after that? Or would that change the expression?