What does $\lim_{\theta \uparrow 1}$ mean in criticality of Branching process?

Question

What does $\lim_{\theta \uparrow 1}$ mean in criticality of Branching process?

Assume that $\lim_{\theta \uparrow 1} F'(\theta)=F'(1) < \infty$.

Where $F$ is is prob. generating function:

$$F(\theta):= \sum_{l=0}^{\infty} \theta^l p_l$$

Answer 1

It is simply the left hand limit of $F'(\theta)$ as $\theta \to 1$. You take limit through values of $\theta <1$.