Differential privacy provides a rigorous framework to quantify data privacy, and has received considerable interest recently. A randomized mechanism satisfying $(ε, δ)$-differential privacy (DP) roughly means that, except with a small probability $δ$, altering a record in a dataset cannot change the probability that an output is seen by more than a multiplicative factor $e^ε $. A well-known solution to $(ε, δ)$-DP is the Gaussian mechanism initiated by Dwork et al. [1] in 2006 with an improvement by Dwork and Roth [2] in 2014, where a Gaussian noise amount $\sqrt{2\ln \frac{2}δ} \times \fracΔε$ of [1] or $\sqrt{2\ln \frac{1.25}δ} \times \fracΔε$ of [2] is added independently to each dimension of the query result, for a query with $\ell_2$-sensitivity $Δ$. Although both classical Gaussian mechanisms [1,2] assume $0 < ε\leq 1$, our review finds that many studies in the literature have used the classical Gaussian mechanisms under values of $ε$ and $δ$ where the added noise amounts of [1,2] do not achieve $(ε,δ)$-DP. We obtain such result by analyzing the optimal noise amount $σ_{DP-OPT}$ for $(ε,δ)$-DP and identifying $ε$ and $δ$ where the noise amounts of classical mechanisms are even less than $σ_{DP-OPT}$. Since $σ_{DP-OPT}$ has no closed-form expression and needs to be approximated in an iterative manner, we propose Gaussian mechanisms by deriving closed-form upper bounds for $σ_{DP-OPT}$. Our mechanisms achieve $(ε,δ)$-DP for any $ε$, while the classical mechanisms [1,2] do not achieve $(ε,δ)$-DP for large $ε$ given $δ$. Moreover, the utilities of our mechanisms improve those of [1,2] and are close to that of the optimal yet more computationally expensive Gaussian mechanism.