  For a transitive infinite connected graph $G$, let $\mu(G)$ be its connective
constant. Denote by $\mathbf{\cal G}$ the set of Cayley graphs for finitely
generated infinite groups with an infinite-order generator which is independent
of other generators. Assume $G\in\mathbf{\cal G}$ is a Cayley graph of a
finitely presented group, and Cayley graph sequence
$\{G_n\}_{n=1}^{\infty}\subset \mathbf{\cal G}$ converges locally to $G.$ Then
$\mu(G_n)$ converges to $\mu(G)$ as $n\rightarrow\infty.$ This confirms
partially a conjecture raised by Benjamini [2013. {\it Coarse geometry and
randomness.} Lect. Notes Math. {\bf 2100}. Springer.] that connective constant
is continuous with respect to local convergence of infinite transitive
connected graphs.
