Locally finite graph without an optimal path
It can happen. Let $P_0,P_1,\ldots$ be a one-way infinite path, where $P_0$ is the start node, and the edge from $P_i$ to $P_{i+1}$ has weight $1/2^{i+1}$. Let $Q_0,Q_1,\ldots$ be a one-way infinite path, where $Q_0$ is the goal node, and the edge from $Q_i$ to $Q_{i+1}$ has weight $1/2^{i+1}$. Connect $P_i$ to $Q_i$ with an edge of weight $1/2^{i-2}$. The weight of the path $P_0-\cdots-P_i-Q_i-\cdots-Q_0$ is
$$(1/2+\cdots+1/2^i)+1/2^{i-2}+(1/2^i+\cdots+1/2) = \\
(1+\cdots+1/2^{i-1})+1/2^{i-2} = \\
(2-1/2^{i-1})+1/2^{i-2} = \\
2+1/2^{i-1} \, .
$$ So for each $\epsilon &gt; 0$ there is a $P-Q$ path of weight $2+\epsilon$, but there is no path of weight $2$.