Compute $\lim_{t\searrow 0}\frac{F(x,t)-f(x)}{t}$ where $F(x,t)$ is a complicated integral. Consider a function $F(x,t)$ defined by $$F(x,t)=\frac{1}{\sqrt{2\pi (1-e^{-2t})}}\int_{\mathbb{R}}\exp\left(-\frac{|y-xe^{-t}|^{2}}{2(1-e^{-2t})}\right)f(y)\,dy.$$ Then I want to compute the limit $$\lim_{t\searrow 0}\frac{F(x,t)-f(x)}{t}.$$
It is undecided and actually part of the question what property of $f(x)$ should have, so that the above limit converges uniformly (or at least pointwise).
For now, I guess $f(y)$ should at last be uniformly continuous on a compact support, perhaps it should be $2-$times continuously differentiable, but I am not sure.
Then, I try to compute the limit, but it turned out that I don't really know how to start. We can definitely analyze the thing without taking the limit: $$\frac{F(x,t)-f(x)}{t}=\frac{1}{t\sqrt{2\pi (1-e^{-2t})}}\int_{\mathbb{R}}\exp\left(-\frac{|y-xe^{-t}|^{2}}{2(1-e^{-2t})}\right)f(y)\,dy-\frac{f(x)}{t}.$$
Since we have really complicated things in the integral, my expectation is that the first term just converges to $0$ uniformly. 
By the assumption that $f(y)$ is bounded uniformly continuous, we can bound the integral with sup norm, $$\left|\int_{\mathbb{R}}\exp\left(-\frac{|y-xe^{-t}|^{2}}{2(1-e^{-2t})}\right)f(y)\,dy\right|\leq \|f\|_{\infty}\int_{\mathbb{R}}\left|\exp\left(-\frac{|y-xe^{-t}|^{2}}{2(1-e^{-2t})}\right)\right|dy,$$ but then I don't know how to proceed...
Is there anyway to compute such a limit? Thank you!
 A: This is a suggestion.
The function 
\begin{aligned}
u(x;t)=\frac{1}{\sqrt{2\pi t}}\int_\mathbb{R}e^{-\frac{|y-x|^2}{2t}}f(y)\,dy=W_t*f(x)
\end{aligned}
where $W_t(y)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{y^2}{2t}}$ is solution to the heat equation problem with initial condition $u(x,0)=f(x)$
\begin{aligned}
\partial_t u&=\frac12\partial^2_{xx}u\\
u(x,0)&=f(x)
\end{aligned}
$f$ in principle can be in $L_1$. Assuming continuity also gives $u(x,t)\rightarrow f(x)$ as $t\rightarrow0$.
What you have is 
$$F(x,t)=u(xe^{-t},1- e^{-2t})$$
So twice differentiable might be a reasonable extra assumption. The problem can also be posed in terms of the generator of Brwonian motion, which gives you the heat equation.
As $F(x,0)=u(x,0)=f(x)$
\begin{aligned}
\lim_{t\rightarrow0}\frac{F(x,t)-f(x)}{t}&=(\partial_tF)(x,t)|_{t=0}\\
&= (\partial_xu(xe^{-t},1-e^{-2t}),\partial_tu(xe^{-t},1-e^{-2t}))\cdot(-xe^{-t},2e^{-2t})|_{t=0}\\
&=-x\partial_xu(x,0)+2\partial_tu(x,0)=-xf'(x)+f''(x)
\end{aligned}under the assumption that $f$ is twice differentiable in $L_1$, e.g., if $f$ is a Schwartz function.
