Limiting fraction of association
Start from 

$$K_\mathrm{b} = \frac{\ce{[HB+]}\ce{[OH-]}}{\ce{[B]}}$$

Define 

$$\alpha = \frac{\ce{[HB+]}}{\ce{[B]} + \ce{[HB+]}} $$

or

$$\frac{\ce{[HB+]}}{\ce{[B]}}=\frac{\alpha}{1-\alpha}$$

so that 

$$K_\mathrm{b} = \frac{\alpha\ce{[OH-]}}{1-\alpha}$$

or 

$$\alpha = \frac{K_\mathrm{b}}{K_\mathrm{b}+\ce{[OH-]}}$$

But in the limit of infinite dilution the concentration of $\ce{OH^-}$ is solely determined by the autodissociation of water (if the solution is pure unbuffered water) and

$$\ce{[OH-]} = \sqrt{K_\mathrm{w}}$$

so that 

$$\alpha = \frac{K_\mathrm{b}}{K_\mathrm{b}+K_\mathrm{w}^{1/2}}$$

But at RTP $K_\mathrm{w}^{1/2}\approx 10^{-7}$ resulting in the final expression

$$\alpha = \frac{10^{7}K_\mathrm{b}}{10^{7}K_\mathrm{b}+1}$$