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) $4 \mathbf { i }$

Question

Quizplus · Accepted Answer

We can find the limit by taking the limit of each component of the vector separately.

$$\lim _ { t \rightarrow 0 } \left( t ^ { 2 } + 3 \right) \mathbf { i } = 3 \mathbf { i }$$
$$\lim _ { t \rightarrow 0 } \cos 5t \mathbf { j } = \cos 0 \mathbf { j } = 1\mathbf { j }$$
$$\lim _ { t \rightarrow 0 } -6 \mathbf { k } = -6 \mathbf { k }$$

So the limit of the entire vector is $\boxed{\mathrm{(C)}\ 3\mathbf{i}+\mathbf{j}-6\mathbf{k}}$.

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