Vector Functions — Question 1

Question 1 – Solution

Strategy Intersect the domains of all three scalar components; vector limits exist exactly when every component limit is finite.

Domain The restrictions are $t\ge-2$, $t<1$, and $t\ne\pm 1$. Hence $$\boxed{[-2,-1)\cup(-1,1)}.$$ Also $\vec r(-1/2)=\left\langle\sqrt{3/2},\ln(3/2),-4/3\right\rangle$.

Limit As $t\to-1^+$, $1/(t^2-1)\to-\infty$, so no finite vector limit exists in $\mathbb R^3$.