I solved the distributional limit as seen below
$$\lim_{x\rightarrow+0}\left(\int_0^\infty \exp(\frac{-ARx}{2})\cos(R(y-t)) \, dR\right)=\frac{2\pi \delta(t-y)}{A}$$
where $ -1<y<1 $ , $ -1<t<1 $ and $A$ is a positive constant.
I am not sure about the existence of the multiplier $\frac{2}{A}$ on the rigth hand side of the equation. Does it exist? if it does not, what is the correct solution.
Best wishes..
Rewrite your integral as $$ I(A,t,y)=\lim_{x\rightarrow0+}\Re\int_{0}^{\infty}e^{-ARx/2} e^{iR(y-t)} =\lim_{x\rightarrow0+}\Re\frac{1}{Ax/2-i(y-t)}=\lim_{x\rightarrow0+}\frac{Ax/2}{(Ax/2)^2+(y-t)^2} $$
It's well known that such expressions are a representation of Dirac's Delta distribution: $\delta(y)=\lim_{x\rightarrow0}\frac{x}{x^2+y^2}$
We obtain $$ I(A,t,y)=\frac{2 \pi}{A}\delta\left(\frac{4}{A^2}(y-t)\right) $$
Using the identiy $\delta(ax)=\frac{1}{|a|}\delta(x)$ this simplifies to $$ I(A,t,y)=\frac{A\pi}{2}\delta\left(y-t\right) $$ So i think your solution is not correct.