Find the limit.
-$$\mathbf { r } ( \mathrm { t } ) = \left( 1 - \mathrm { t } ^ { 2 } \right) \mathbf { j } + 3 \mathrm { tk } ; - 1 \leq \mathrm { t } \leq 1$$
A) Figure 2
B) Figure 8
C) Figure 4
D) Figure 7

Question

Quizplus · Accepted Answer

The path traced by $\mathbf{r}(\mathrm{t})$ is a helix with axis along the $\mathbf{j}$ direction. As $\mathrm{t}$ approaches $\pm \infty$, $\mathbf{r}(\mathrm{t})$ spirals along the helix and does not approach any finite point. However, as $\mathrm{t}$ approaches $\pm 1$, $\mathbf{r}(\mathrm{t})$ approaches the point $(0,\pm 3,0)$ on the $\mathbf{j}$ axis. Looking at the answer choices, only Figure 7 depicts a helix with axis along the $\mathbf{j}$ direction and approaches the points $(0,3,0)$ and $(0,-3,0)$ as $\mathrm{t}$ approaches $\pm 1$. Thus, the answer is D.

Find the Limit r(t)=(1−t2)j+3tk;−1≤t≤1\mathbf { r } ( \mathrm { t } ) = \left( 1 - \mathrm { t } ^ { 2 } \right) \mathbf { j } + 3 \mathrm { tk } ; - 1 \leq \mathrm { t } \leq 1r(t)=(1−t2)j+3tk;−1≤t≤1

Find the Limit $\mathbf { r } ( \mathrm { t } ) = \left( 1 - \mathrm { t } ^ { 2 } \right) \mathbf { j } + 3 \mathrm { tk } ; - 1 \leq \mathrm { t } \leq 1$