= Solution
{c}
The integrated friction is exactly
$$
\widetilde\kappa_c(0)=\int_0^\infty ce^{-ct}\,dt=1.
$$
For every finite $c>0$, the roots of $s^2+cs+c$ have negative real parts. The <Green-Kubo formula for diffusion> therefore gives the <diffusion coefficient> directly:
$$
D_c=\widetilde\chi_c(0)=\frac5{\widetilde\kappa_c(0)}=5.
$$
Consequently
$$
\boxed{\lim_{c_1\to\infty}D=5.}
$$
Integrating the limiting <velocity> <autocorrelation function> gives the same value, $\int_0^\infty5e^{-\tau}\,d\tau=5$. Computing $D_c$ before taking the limit also avoids having to justify an interchange of the infinite-time <integral> and the parameter limit.
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