Calculate the limit of a vector?

Question

Find $$~\lim_{t→0} f (t)~$$ if it exists

$$f(t) = \dfrac{\sin t}{2t} ~\hat i + \, e^{2 t}~\hat j + \dfrac{t^2}{e^{t}}~\hat k ~.$$

When I plug in $~t\to 0~$ for $~f(t)~$, I get $0+j+0$.
But the answer is $\left( \frac{1}{2} ~\hat i + ~\hat j\right)$.
Could someone tell me where I have made grievous mistake?

Answer 1

Note that $$ \lim_{t\to0}\frac{\sin t}{2t} = \frac{1}{2}. $$ (Think L'Hopital's!)

Answer 2

You should use L'Hospital Rule for $i$ component only, getting $\frac12$. For $j$ component it goes to $1 $, and $k$ component limit is clearly zero.