Evaluate the limit of the following expression:
\[\lim_{t\to0}\left(\frac{1-\cos(t)}{t}\right)\vec{i}+7t^2\vec{j}+e^{-t}\vec{k}\]
Enter your answer in the form of \(\vec{a}\vec{i}+\vec{b}\vec{j}+\vec{c}\vec{k}\), where \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are the components of the vector.
Which of the following options represents the correct answer?

A) \(\vec{1}\vec{i}+0\vec{j}+1\vec{k}\)
B) \(\vec{0}\vec{i}+7\vec{j}+1\vec{k}\)
C) \(\vec{1}\vec{i}+7\vec{j}+0\vec{k}\)
D) \(\vec{0}\vec{i}+0\vec{j}+0\vec{k}\)