Find $\lim _ { t \rightarrow \infty } \left[ \frac { t + 1 } { t - 1 } \mathbf { i } + \left( 1 + \frac { 1 } { t } \right) ^ { t } \mathbf { j } + \arctan t \mathbf { k } \right]$
A)i + ej + $\frac { \pi } { 4 }$ k
B)i + ej + $\frac { \pi } { 2 }$ k
C)i + $\frac { \pi } { 4 }$ k
D)i + $\frac { \pi } { 2 }$ k
E)3i + ej + $\frac { \pi } { 4 }$ k
F)3i + ej + $\frac { \pi } { 2 }$ k
G)3i + $\frac { \pi } { 4 }$ k
H)3i + $\frac { \pi } { 2 }$ k

Question

Find $\lim _ { t \rightarrow \infty } \left[ \frac { t + 1 } { t - 1 } \mathbf { i } + \left( 1 + \frac { 1 } { t } \right) ^ { t } \mathbf { j } + \arctan t \mathbf { k } \right]$ 
A)i + ej + $\frac { \pi } { 4 }$ k
B)i + ej + $\frac { \pi } { 2 }$ k
C)i + $\frac { \pi } { 4 }$ k
D)i + $\frac { \pi } { 2 }$ k
E)3i + ej + $\frac { \pi } { 4 }$ k
F)3i + ej + $\frac { \pi } { 2 }$ k
G)3i + $\frac { \pi } { 4 }$ k
H)3i + $\frac { \pi } { 2 }$ k

Quizplus · Accepted Answer

The limit of the first term as $t \rightarrow \infty$ is $\lim _ { t \rightarrow \infty } \frac { t + 1 } { t - 1 } = 1$, so it simplifies to $\mathbf { i }$. The second term is a well-known limit, $\lim _ { t \rightarrow \infty } \left( 1 + \frac { 1 } { t } \right) ^ { t } = e$, which simplifies to $e\mathbf { j }$. The third term, $\lim _ { t \rightarrow \infty } \arctan(t)$, approaches $\frac { \pi } { 2 }$ as $t$ goes to infinity, resulting in $\frac { \pi } { 2 }\mathbf { k }$. Therefore, the correct answer is $i + e\mathbf { j } + \frac { \pi } { 2 }\mathbf { k }$.

Find A)i + Ej π4\frac { \pi } { 4 }4π​

Find A)i + Ej $\frac { \pi } { 4 }$