Vector Valued Functions

1. Find the following limit:  $\displaystyle{\lim_{t \to e^2} \left \langle t\ln(t), \frac{\ln t}{t^2}, \sqrt{\ln(t^2)}\right \rangle}$    solution

2. Find the domain of the vector-valued function.   ${\bf r}(t) = \langle t^2, \tan t, \ln t \rangle$    solution

3. Eliminate the parameter t, write the equation in Cartesian coordinates, then sketch the graphs of the vector-valued functions.    solution

(a)  ${\bf r}(t) = t^3 \, {\bf i} + 2t \, {\bf j}$             (b)  ${\bf r}(t) = 2 \cos t \, {\bf i} + 2 \sin t \,  {\bf j}$