Let $f$ be integrable with respect to a Lebesgue measure. Evaluate the limit, $$\lim_{n \to \infty} \int_{-\infty}^{\infty} f(x-n)\left(\frac{1}{1+|x|}\right)\,dx$$
I tried change of variables but I don't know what to do after that.
2026-04-23 16:03:38.1776960218
Lebesgue Integration Question
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$$\int_{-\infty}^{+\infty}f(x-n)\left(\frac{1}{1+|x|}\right)\,dx = \int_{-\infty}^{+\infty}f(x)\cdot\frac{dx}{1+|x+n|}$$ so consider the sequence of functions given by $g_n(x)=f(x)\cdot \frac{1}{1+|x+n|}$. They are dominated by an integrable function $f(x)$ and almost everywhere pointwise convergent to zero, so the limit is zero.