  We study Bernoulli first-passage percolation (FPP) on the triangular lattice
$\mathbb{T}$ in which sites have 0 and 1 passage times with probability $p$ and
$1-p$, respectively. Denote by $\mathcal {C}_{\infty}$ the infinite cluster
with 0-time sites when $p>p_c$, where $p_c=1/2$ is the critical probability.
Denote by $T(0,\mathcal {C}_{\infty})$ the passage time from the origin 0 to
$\mathcal {C}_{\infty}$. First we obtain explicit limit theorem for
$T(0,\mathcal {C}_{\infty})$ as $p\searrow p_c$. The proof relies on the limit
theorem in the critical case, the critical exponent for correlation length and
Kesten's scaling relations. Next, for the usual point-to-point passage time
$a_{0,n}$ in the critical case, we construct subsequences of sites with
different growth rate along the axis. The main tool involves the large
deviation estimates on the nesting of CLE$_6$ loops derived by Miller, Watson
and Wilson (2016). Finally, we apply the limit theorem for critical Bernoulli
FPP to a random graph called cluster graph, obtaining explicit strong law of
large numbers for graph distance.
