I am investigating the limit
limx→∞xtan−1(f(x)x+g(x))
given that f(x)→0 and g(x)→0 as x→∞. My initial guess is the limit exists since the decline rate of tan−1 will compensate the linearly increasing x. But I'm not sure if the limit can be non zero. My second guess is the limit will always zero but I can't prove it. Thank you.
EDIT 1: this problem ca be reduced into proving that limx→∞xtan−1(M/x)=M for any M∈R. Which I cannot prove it yet.
EDIT 2: indeed limx→∞xtan−1(M/x)=M for any M∈R. Observe that
limx→∞xtan−1(M/x)=limx→0tan−1(Mx)x.
By using L'Hopital's rule, the right hand side gives M. So the limit which is being investigated is equal to zero for any f(x) and g(x) as long as f(x)→0 and g(x)→0 as x→∞. The problem is solved.
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