Recent advance in diffusion models incorporates the Stochastic Differential Equation (SDE), which brings the state-of-the art performance on image generation tasks. This paper improves such diffusion models by analyzing the model at the zero diffusion time. In real datasets, the score function diverges as the diffusion time ($t$) decreases to zero, and this observation leads an argument that the score estimation fails at $t=0$ with any neural network structure. Subsequently, we introduce Unbounded Diffusion Model (UDM) that resolves the score diverging problem with an easily applicable modification to any diffusion models. Additionally, we introduce a new SDE that overcomes the theoretic and practical limitations of Variance Exploding SDE. On top of that, the introduced Soft Truncation method improves the sample quality by mitigating the loss scale issue that happens at $t=0$. We further provide a theoretic result of the proposed method to uncover the behind mechanism of the diffusion models.