$"$Let $f:\mathbb R^n\rightarrow\mathbb R$ be differentiable... The directional derivative of $f$ at the point $x$ in the direction $h$ is given by $Df(x,h)=\lim_{t\rightarrow 0}\frac 1 t(f(x+th)-f(x))$ and satisfies $Df(x,h)=\boldsymbol \nabla f(x)^Th$.
In general, $f(x+h)-f(x)\ge\frac{f(x+2^{-k}h)-f(x)}{2^{-k}}$ for all $k\in\mathbb N$.
Thus taking the limit $k\rightarrow\infty$ we see that $$f(x+h)-f(x)\ge \limsup_{k\rightarrow\infty}\frac{f(x+2^{-k}h)-f(x)}{2^{-k}}=Df(x,h)=\boldsymbol \nabla f(x)^Th "$$
Is there any reason they took the limsup and not the regular limit? does it mean the same thing here