Tuesday, November 12, 2019

calculus - Regarding limits: limlimitsntoinftyleft(fracfleft(x+frac1nright)f(x)right)n



If f is positive and differentiable in (0,), then I want to find the following limit.



limn(f(x+1n)f(x))n.



I have done as follows:
limn(f(x+1n)f(x))n=(limnf(x+1n)f(x))n=1 as f is continuous at x=1n. Am I right? I doubt. Please help!


Answer




log[(f(x+1n)f(x))n]=nlogf(x)+1nf(x)+ϵ(1n)1nf(x)=nlog[1+1nf(x)f(x)+ϵ(1n)1n]


with lim0ϵ=0.
n1nf(x)f(x)=f(x)f(x)
so the limit is expf(x)f(x)


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