  We consider a simultaneous small noise limit for a singularly perturbed
coupled diffusion described by \begin{eqnarray*} dX^{\varepsilon}_t &=&
b(X^{\varepsilon}_t, Y^{\varepsilon}_t)dt + \varepsilon^{\alpha}dB_t,
  dY^{\varepsilon}_t &=& - \frac{1}{\varepsilon} \nabla_yU(X^{\varepsilon}_t,
Y^{\varepsilon}_t)dt + \frac{s(\varepsilon)}{\sqrt{\varepsilon}} dW_t,
\end{eqnarray*} where $B_t, W_t$ are independent Brownian motions on ${\mathbb
R}^d$ and ${\mathbb R}^m$ respectively, $b : \mathbb{R}^d \times \mathbb{R}^m
\rightarrow \mathbb{R}^d$, $U : \mathbb{R}^d \times \mathbb{R}^m \rightarrow
\mathbb{R}$ and $s :(0,\infty) \rightarrow (0,\infty)$. We impose regularity
assumptions on $b$, $U$ and let $0 < \alpha < 1.$ When $s(\varepsilon)$ goes to
zero slower than a prescribed rate as $\varepsilon \rightarrow 0$, we
characterize all weak limit points of $X^{\varepsilon}$, as $\varepsilon
\rightarrow 0$, as solutions to a differential equation driven by a measurable
vector field. Under an additional assumption on the behaviour of $U(x, \cdot)$
at its global minima we characterize all limit points as Filippov solutions to
the differential equation.
