Activity 10.3.2

For each of the following vector-valued functions, state any values of $a$ for which $\displaystyle\lim_{t\to a}\vr(t)$ will not exist and find $\vr\, '(t)$.

1. $\vr(t) = \langle \cos(t), t\sin(t), \ln(t) \rangle$.

2. $\vr(t) = \langle t^2 + 3t, e^{-2t}, \displaystyle\frac{t}{t^2 + 1} \rangle$.

3. $\vr(t) = \langle \tan(t), \cos(t^2), te^{-t} \rangle$.

4. $\vr(t) = \left\langle \sqrt{t^4 + 4}, \frac{2}{t^2+t} , e^{2t} \sin(-2t) \right\rangle$.