# How to integrate in Real analysis?

Let $f \in L^1(-\infty, \infty)$. How to find the limit: $$\lim_{n \to \infty} \int_{-\infty}^{\infty} \frac{f(x)e^{nx} dx}{1+e^{nx}}.$$ What are the difference between the integration in Riemann integral and Lebesgue integral when we compute integrations? Thank you very much.

-

Hint 2: Prove the following equality $$\lim\limits_{n\to\infty}\frac{f(x)e^{nx}}{1+e^{nx}}= \begin{cases} f(x)&\quad\text{ if }\quad x>0\\ 0.5 f(0)&\quad\text{ if }\quad x=0\\ 0&\quad\text{ if }\quad x<0\\ \end{cases}$$
For a function $f$ put $f+ = f\vee 0$ and $f^- = (-f)\vee 0$. Then $f = f^+ - f^-$. If $\int f^+$ and $\int f^-$ are finite and $f$ is a.e. continuous, the Riemann and Lebesgue integrals agree. In particular, if $f$ is absolutely integrable in the Riemann sense, it agrees with the Lebesgue integral.