Final answer:
The limit as t approaches 0 for the given vector does not exist for the first component due to division by zero, is 0 for the second component as identical terms cancel out, and is 6 for the third component.
Step-by-step explanation:
The limit in question is lim (t → 0) [2et - 2/t, 1/t - 1/t, 6/(1 + t)]. To find this limit, we evaluate each component of the vector separately.
For the first component, 2et - 2/t, as t approaches 0, the exponential function et approaches 1. Therefore, the first component approaches 2*1 - 2/0, which is undefined due to division by zero.
The second component, 1/t - 1/t, simplifies to 0 since identical terms cancel each other out.
The third component, 6/(1 + t), as t approaches 0, simplifies to 6/1, which equals 6.
So, the limit of the given vector as t approaches 0 does not exist for the first component, is 0 for the second component, and is 6 for the third component.