real analysis - How to find $lim_{ntoinfty}int_{1/n}^{2/n}frac{f(x)}{x},mathrm{d}x$?

Suppose $f:\mathbf{R}\to\mathbf{R}$ is continuous and $f(0)=3$. Find

$$\lim_{n\to\infty}\int_{1/n}^{2/n}\frac{f(x)}{x}\mathrm{d}x.$$

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.)