Differential private stochastic gradient descent (DP-SGD) with gradient clipping (DP-SGD-GC) is an effective optimization algorithm that can train machine learning models with a privacy guarantee. Despite the popularity of DP-SGD-GC, its convergence in unbounded domain without the Lipschitz continuous assumption is less-understood; existing analysis of DP-SGD-GC either impose additional assumptions or end up with an utility bound that involves an non-vanishing bias term. In this work, for smooth and unconstrained problems, we improve the current analysis and show that DP-SGD-GC can achieve a vanishing utility bound without any bias term. Furthermore, when the noise generated from subsampled gradients is light-tailed, we prove that DP-SGD-GC can achieve nearly the same utility bound as DP-SGD applies to the Lipschitz continuous objectives. As a by-product, we propose a new clipping technique, called value clipping, to mitigate the computational overhead caused by the classic gradient clipping. Experiments on standard benchmark datasets are conducted to support our analysis.