= Solution
Use one $d$-dimensional <Brownian motion> $W$ to couple the <diffusion processes>
$$
X_t^\varepsilon=x_0+\int_0^t b(X_s^\varepsilon)ds+\varepsilon W_t.
$$
The <global existence theorem for stochastic differential equations with Lipschitz coefficients> supplies a unique nonexplosive <strong stochastic solution>. Let $x$ be the deterministic drift trajectory. Subtraction and the <Gronwall inequality> give, on every finite horizon $t$,
$$
\boxed{\sup_{s\le t}|X_s^\varepsilon-x_s|
\le\varepsilon e^{Lt}\sup_{s\le t}|W_s|\longrightarrow0\quad\text{almost surely}.}
$$
This also follows directly from the integral equations with continuous forcing, so the coupling can be chosen simultaneously for all $\varepsilon$.
For completeness derive the needed <Feynman-Kac formula> with its potential sign. For fixed $t$, apply the <Itô formula> and the <Itô product rule> to
$$
Y_s=\exp\left(\int_0^s c(X_v^\varepsilon)dv\right)
u^\varepsilon(t-s,X_s^\varepsilon),\qquad 0\le s\le t.
$$
Its drift is the exponential factor times $-u_t^\varepsilon+(\varepsilon^2\Delta/2+b\cdot\nabla)u^\varepsilon+c u^\varepsilon$, which vanishes by the <partial differential equation>. It is a <local martingale>. Boundedness of $u^\varepsilon$ on the finite time slab and boundedness of $c$ make $Y$ bounded there, so it is a true <martingale>. Its endpoint expectations give
$$
u^\varepsilon(t,x_0)=\mathbb E\left[f(X_t^\varepsilon)
\exp\left(\int_0^t c(X_s^\varepsilon)ds\right)\right].
$$
The deterministic trajectory on $[0,t]$ is compact. Uniform pathwise convergence puts all sufficiently small-noise paths in a fixed compact neighbourhood of that trajectory. Continuity of $c$ therefore implies $\sup_{s\le t}|c(X_s^\varepsilon)-c(x_s)|\to0$; global <uniform continuity> of $c$ is not required. Continuity of $f$ gives convergence of the terminal factor. Finally,
$$
\left|f(X_t^\varepsilon)\exp\left(\int_0^tc(X_s^\varepsilon)ds\right)\right|
\le\|f\|_\infty e^{t\|c\|_\infty}.
$$
The <dominated convergence theorem> proves the <zero-noise limit with a bounded potential>:
$$
\boxed{u^\varepsilon(t,x_0)\longrightarrow
f(x_t)\exp\left(\int_0^t c(x_s)ds\right).}
$$
At $t=0$ this is the given initial condition. All boundedness arguments concern a fixed finite horizon; no uniform bound over infinite time is needed.
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