Inside $S^{1}_{2\pi}\times\mathbb{R}$ with the standard Euclidean metric, take $L_{k}$ as the graph over $S^{1}$ of the function $\frac{1}{k}\sin(kx)$. Then $L_{k}$ are Lagrangian currents, which converge to $S^{1}$ as currents, but due to the high oscillation, $\liminf Mass(L_{k})>Mass(S^{1})$, and $L_{k}$ do not converge to $S^{1}$ in the varifold sense. The Lagrangian angle of $L_{k}$ is prescribed by $\tan\theta=\cos(kx)$, which converges to zero in the current sense, but not strongly in $L^{1}$.