Dirac delta distribution

Question

when I want to show that limit of the following function $$f_\epsilon(x)=\frac{1}{2\epsilon}e^{-\frac{|x|}{\epsilon}}~,$$ namely, $$\lim_{\epsilon\to 0} f_\epsilon(x)$$ is a presentation of $\delta$-distribution. And for that, I have to show that for each test function $g$ we have: $$\int_{-\infty}^{\infty}dx\lim_{\epsilon\to 0} f_\epsilon(x)g(x)=g(0)$$ We can assume to interchange integration and limits. How do I do that? Would be very thankful for help!

Answer 1

This means do the integral of $f_{\epsilon}(x)g(x)$ for a given $g()$, then take the limit as $\epsilon \to 0$ of the answer that you get.

Note: to do the integral, you should split up into $[0,\infty)$ and $(-\infty, 0)$ to tackle the modulus.