a7db81…: “Through the winter cube”

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Consider the vector-valued function $\vec {p}(t) = \vector {5t, \frac {e^{2t}-1}{t}, t^3-4t}$ Find the domain of $\vec {p}$ . $D = \left (\answer {-\infty },\answer {0}\right ) \cup \left (\answer {0},\answer {\infty }\right )$

Find the intervals where $\vec {p}$ is continuous: $\left (\answer {-\infty },\answer {0}\right ) \cup \left (\answer {0},\answer {\infty }\right )$

Define a new vector-valued function $\vec {r}$ such that $\vec {r}(t) = \vec {p}(t) \quad t\ne 0$ and $\vec {r}(0) = \vector {0,a,0}$ . Find the value of $a$ such that $\vec {r}$ is continuous at $t=0$ . $a = \answer {2}$