Find the limit of the vector as $~t~$ approaches $~0~$? Here's the vector: $$e^{-6t}~\vec i + \frac{t^2}{\sin^2t}~\vec j + \sin(6t)~\vec k$$
Don't I just take the limit as $t$ approaches $0$ from each individual component? 
As a result, I got $~\vec i + \vec j~$, which was apparently incorrect.

$$\lim_{t\to 0} \left\{e^{-6t}~\vec i + \frac{t^2}{\sin^2t}~\vec j + \sin(6t)~\vec k\right\}$$
$$=\left\{\lim_{t\to 0}  ~e^{-6t}~\right\}~\vec i + \left\{\lim_{t\to 0}  ~\frac{t^2}{\sin^2t}~\right\}~\vec j + \left\{\lim_{t\to 0}  ~ \sin(6t)~\right\}~\vec k $$
$$=e^0~\vec i ~+~\vec j~+~\sin(0)~\vec k$$
$$=\vec i ~+~\vec j$$



*

*$$~\lim_{t\to 0}  ~\frac{t^2}{\sin^2t}=~\lim_{t\to 0}  \left(~\frac{t}{\sin t}\right)^2=\left(~\lim_{t\to 0}~\frac{t}{\sin t}\right)^2=\frac{1}{\lim_{t\to 0}~\frac{\sin t}{t}}=1~$$
