Find the limit $\lim _ { t \rightarrow 0 } \left[ \left( t ^ { 2 } + 3 \right) \mathbf { i } + \cos 5 t \mathbf { j } - 6 \mathbf { k } \right]$ .
A) $3 \mathbf { i } - 6 \mathbf { k }$
B) $4 \mathbf { i } + \mathbf { j }$
C) $3 \mathbf { i } + \mathbf { j } - 6 \mathbf { k }$
D) $4 i$

Question

Quizplus · Accepted Answer

As $t$ approaches $0$, $t^2$ approaches $0$. Therefore, the first component of the vector approaches $3$, and the third component approaches $-6$. Since $\cos 5t$ is bounded between $-1$ and $1$, the second component remains bounded as $t$ approaches $0$. Therefore, the limit of the vector is $\boxed{3 \mathbf{i} + \mathbf{j} - 6 \mathbf{k}}$.

Find the Limit lim⁡t→0[(t2+3)i+cos⁡5tj−6k]\lim _ { t \rightarrow 0 } \left[ \left( t ^ { 2 } + 3 \right) \mathbf { i } + \cos 5 t \mathbf { j } - 6 \mathbf { k } \right]limt→0​[(t2+3)i+cos5tj−6k]

Find the Limit $\lim _ { t \rightarrow 0 } \left[ \left( t ^ { 2 } + 3 \right) \mathbf { i } + \cos 5 t \mathbf { j } - 6 \mathbf { k } \right]$