A Particle Moves with Position Vector $\vec { r } ( t ) = \ln t \vec { i } + t ^ { - 1 } \vec { j } + e ^ { - t } \vec { k }$

Multiple Choice

A particle moves with position vector $\vec { r } ( t ) = \ln t \vec { i } + t ^ { - 1 } \vec { j } + e ^ { - t } \vec { k }$ . Describe the movement of the particle as t $\rightarrow$ $\infty$ .

A) The particle will approach the positive x-axis asymptotically as t $\rightarrow$ $\infty$ .Also, since each component of $\vec { v } ( t )$ approaches 0 as t $\rightarrow$ $\infty$ , we expect the particle to approach the positive y-axis with slower and slower speed.
B) The particle will approach the positive y-axis asymptotically as t $\rightarrow$$\infty$ .Also, since each component of $\vec { v } ( t )$ approaches 0 as t $\rightarrow$ $\infty$ , we expect the particle to approach the positive x-axis with slower and slower speed.
C) The particle will approach the positive x-axis asymptotically as t $\rightarrow$ $\infty$ .Also, since each component of $\vec { v } ( t )$ approaches 0 as t $\rightarrow$ $\infty$ , we expect the particle to approach the positive x-axis with increasing speed.
D) The particle will approach the positive x-axis asymptotically as t $\rightarrow$$\infty$ .Also, since each component of $\vec { v } ( t )$ approaches 0 as t $\rightarrow$ $\infty$ , we expect the particle to approach the positive x-axis with slower and slower speed.
E) The particle will approach the positive x-axis asymptotically as t $\rightarrow$ $\infty$ .Also, since each component of $\vec { v } ( t )$ approaches 0 as t $\rightarrow$ $\infty$ , we expect the particle to approach the positive z-axis with slower and slower speed.