For every monopole point $p_{m}\in\mathcal{P}_{m_{0}}$ and for every fixed positive constant $\sigma>0$, we have the following $C^{\infty}$-convergence

(7.52)

$$(\mathcal{N}_{m_{0}}^{4},\tilde{g}_{\sigma,\beta},p_{m})\xrightarrow{C^{\infty}}(\mathbb{R}^{4},\tilde{g}_{\sigma,\infty},\tilde{p}_{m,\infty})\ \text{as}\ \beta\to+\infty,$$

such that $(\mathbb{R}^{4},\tilde{g}_{\sigma,\infty},\tilde{p}_{m,\infty})$ is a Ricci-flat Taub-NUT space with

(7.53)

$$\tilde{g}_{\sigma,\infty}=G_{\sigma}\cdot g_{\mathbb{R}^{3}}+(G_{\sigma})^{-1}\theta^{2}$$

and

(7.54)

$$G_{\sigma}(p)=\frac{1}{2d_{0}(p,0^{3})}+\frac{1}{\sigma^{2}},$$

where $d_{0}$ is the distance function in the Euclidean space $\mathbb{R}^{3}$.