I am studying time independent perturbations in quantum mechanics. In my textbook it states that the first order transition probability from state $a$ to $b$ is proportional to the function
$$F(t, \omega )=\frac{1-cos(\omega t)}{\omega^2} $$
it then remarks that
$$F(t, \omega =0) =t^2/2$$
Can anyone explain why this should be the case?
What it means by $F(t,\omega=0)$ is actually $\lim_{\omega \to 0} F(t,\omega)$. This limit is calculated as follows as follows using L'Hôpital's rule. \begin{align*} &\lim_{\omega \to 0} \frac{1-\cos(\omega t)}{{\omega}^2}\\ &=\lim_{\omega \to 0} \frac{t \sin(\omega t)}{2\omega}\\ &=\lim_{\omega \to 0} \frac{t^2 \cos(\omega t)}{2}\\ &=\frac{t^2}{2} \end{align*}
