S. Weinberg, “The Quantum theory of fields: Foundations” (1995), Eq. 9.2.15

In Weinberg's book The Quantum Theory of Fields, Volume 1 on p.388 (Chapter 9), the following identity is used (with f being any "reasonable" function):

$$f(+\infty) + f(-\infty) = \lim_{\epsilon \rightarrow 0^+} \epsilon \int_{-\infty}^{+\infty} d\tau f(\tau) e^{-\epsilon |\tau|}.\tag{9.2.15}$$

I don't understand the identity in a qualitative / heuristic way.

Answer

I can understand the identity qualitatively now:

If the function f(t) has a well defined value at $t = \pm\infty$, then for large values of $|t|$ the function f is essentially constant with values $f(-\infty)$ for $t < 0$ and $f(+\infty)$ for $t > 0$.

For tiny $\epsilon$ the Exponential factor $e^{-\epsilon |t|}$ is essentially equal to $1$ for $|t| < 1 / \epsilon$, and almost Zero for large $|t| > 1 / \epsilon$.

We then have approximately: $\epsilon\int_{-\infty}^\infty \mathrm{d}t \, f(t) e^{-\epsilon |t|} \approx \epsilon\int_{1 / \epsilon}^0 \mathrm{d}t \, f(-\infty) + \epsilon\int_0^{1 / \epsilon} \mathrm{d}t \, f(+\infty) = f(-\infty) + f(+\infty)$