Given an invertible matrix $M$ which depends on a real parameter $\alpha$. Is the limit $\lim_{\alpha - \alpha_0} \left( M^{-1} \right) $ same as $ \left( \lim_{\alpha - \alpha_0} M\right)^{-1}$, i.e. does the following hold
\begin{equation} \lim_{\alpha - \alpha_0} \left( M^{-1} \right) \overset{\mathrm{?}}{=} \ \left( \lim_{\alpha - \alpha_0} M\right)^{-1} \end{equation} Or is there any relation between these quantities (say one is less or equal to the other)?
The map $M\mapsto M^{-1}$ is continuous on $\mathrm{GL}_n(\mathbb{R})$, see here. Hence the answer is affirmative.