Suppose f:R→R 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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