Answer
$\lim\limits_{t\to1}h(t)$ does not exist.
Work Step by Step
$\lim\limits_{t\to1^-}h(t)=\lim\limits_{t\to1^-}(t^3+1)=(1^-)^3+1=2.$
$\lim\limits_{t\to1^+}h(t)=\lim\limits_{t\to1^+}(\frac{1}{2}(t+1))=\frac{1}{2}(1^++1)=1.$
Since $\lim\limits_{t\to1^+}h(t)\ne\lim\limits_{t\to1^-}h(t)\to\lim\limits_{t\to1}h(t)$ does not exist.
