I have a rather specific question. I constructed a sequence of functions, that is smooth, which I wanted to converge to the characteristic function for $n\to\infty$: $$f^q_n(x)=\frac{1}{1+e^{n[(x-(q-\frac{\varepsilon}{2})(x-(q+\frac{\varepsilon}{2}))]}} $$ In this sequence of functions $q$ is a certain point on the real axis and $\varepsilon>0$. For $q'=q+\varepsilon$ I want to find the limit of $$ \frac{d}{dx}[f^{q'}_n(x)]f^q_n(x)- f^{q'}_n(x)\frac{d}{dx}f^q_n(x)$$ for $n\to\infty$. I tried to sketch it, to get an idea how it might look like:
Because of this I suppose that for $n\to\infty$ it holds that $$ \frac{d}{dx}[f^{q'}_n(x)]f^q_n(x)- f^{q'}_n(x)\frac{d}{dx}f^q_n(x)\to \delta_{q+\frac{\varepsilon}{2}}(x)$$
Do you have any idea how to verify this idea? Thank you very much for any help!