Tuesday, 8 May 2018

real analysis - How to find limntoinftyint2/n1/nfracf(x)x,mathrmdx?




Suppose f:RR is continuous and f(0)=3. Find lim





I don't know how to do this problem, any hints?


Answer



f is continuous at 0, hence there exists \delta>0 such that |f(x)-3|<\delta if |x|<\varepsilon. It follows that the integral L is such that
(3-\delta)\log 2=\int_{1/n}^{2/n}\frac{3-\delta}{x}\mathrm{d}x\le L\le \int_{1/n}^{2/n}\frac{3+\delta}{x}\mathrm{d}x=(3+\delta)\log 2
if n is sufficiently large. Hence the limit is 3\log 2.




(Thanks to Michael Hardy and Winther for spotting a mistake.)


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