In this page of Wikipedia the two notations $$\lim_{t\downarrow 0}\frac{1}{t}\left( T(t)-I\right)x $$ and $$\lim_{t\to 0^+}\left( T(t)-I\right) $$ are used for a map $T : \mathbb{R}^+ \to L(X)$, where $L(X)$ is the set of linear operators in a Banach space.
I don't understand the difference between the two notations. I was convinced that they had the same meaning. Is it the case that in this context there really is a difference?
The symbols $\lim_{t \downarrow 0}$ and $\lim_{t \to 0^+}$ have the same meaning as mentioned in the comments by @angryavian and @Robert Israel. They both the limit as $t$ approaches $0$ from the right.
However, the two limits you mention are different. The first is a limit in the vector space $X$, because $\frac 1t (T(t)-I)x \in X$. On the other hand, the second limit lives in the space $L(X)$, because $T(t)-I \in L(X)$.