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) $3 i - 6 \mathbf { k }$
B) $4 \mathbf { i } + \mathbf { j }$
C) $3 i + j - 6 \mathbf { k }$
D) $4 \mathbf { i }$

Question

Quizplus · Accepted Answer

As $t$ approaches 0, the first term $\left(t^{2}+3\right)\mathbf{i}$ approaches $3\mathbf{i}$. The second term $\cos 5t \mathbf{j}$ oscillates between $\mathbf{j}$ and $-\mathbf{j}$ as $t$ approaches 0, so it does not have a limit. The third term $-6\mathbf{k}$ approaches $-6\mathbf{k}$ as $t$ approaches 0.

Thus, the overall limit becomes $\boxed{3\mathbf{i} - 6\mathbf{k}}$.

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