In this talk, I will explain the following result and outline its proof: Let $X$ be a compact Kähler manifold and $D$ a smooth divisor such that $K_X+D$ is ample. Then the negatively curved Kähler-Einstein metric with cone angle $\beta$ along $D$ converges to the cuspidal Kähler-Einstein metric of Tian-Yau when $\beta$ tends to zero.