\begin{align*} \lim_{x\rightarrow\pm0}\left\{ \Ci\left(\alpha x\right)-\Ci\left(x\right)\right\} & =\lim_{x\rightarrow\pm0}\left\{ -\int_{\alpha x}^{\infty}\frac{\cos t}{t}dt+\int_{x}^{\infty}\frac{\cos t}{t}dt\right\} \\ & =\lim_{x\rightarrow\pm0}\left\{ \int_{x}^{\alpha x}\frac{\cos t}{t}dt\right\} \\ & =\lim_{x\rightarrow\pm0}\left\{ \int_{x}^{\alpha x}\frac{1}{t}\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}}{k!}t^{2k}dt\right\} \\ & =\lim_{x\rightarrow\pm0}\left\{ \int_{x}^{\alpha x}\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}}{k!}t^{2k-1}dt\right\} \\ & =\lim_{x\rightarrow\pm0}\left\{ \int_{x}^{\alpha x}\left(\frac{1}{t}+\sum_{k=1}^{\infty}\frac{\left(-1\right)^{k}}{k!}t^{2k-1}\right)dt\right\} \\ & =\lim_{x\rightarrow\pm0}\left\{ \int_{x}^{\alpha x}\frac{1}{t}dt+\sum_{k=1}^{\infty}\int_{x}^{\alpha x}\frac{\left(-1\right)^{k}}{k!}t^{2k-1}dt\right\} \\ & =\lim_{x\rightarrow\pm0}\left\{ \left(\left[\Log t\right]_{x}^{\alpha x}+\sum_{k=1}^{\infty}\left[\frac{\left(-1\right)^{k}}{k!2k}t^{2k}\right]_{x}^{\alpha x}\right)\right\} \\ & =\lim_{x\rightarrow\pm0}\left(\Log\left(\alpha x\right)-\Log\left(x\right)\right)\\ & =\begin{cases} \Log\alpha & x\rightarrow+0\\ \Log\left(-\alpha\right)-\pi i & x\rightarrow-0 \end{cases} \end{align*}