Evaluate the limit
\[
\lim _{t \rightarrow 0} \frac{\sin (t)}{t} \vec{i}+\ln (t+7) \vec{j}+9 t^{2} \vec{k}
\]
Enter your answer in $a \vec{i}+b \vec{j}+c \vec{k}$ form. However, use the ordinary letters $i, j$, and $k$ for the component basis vectors; you don't need to reproduce the vector arrow notation.

Step 1 ：The limit of a vector is the vector of the limits of its components. So, we can calculate the limit of each component separately.

Step 2 ：For the first component, we have the limit of \(\frac{\sin(t)}{t}\) as \(t\) approaches 0. This is a well-known limit in calculus, and its value is 1.

Step 3 ：For the second component, we have the limit of \(\ln(t+7)\) as \(t\) approaches 0. Since the natural logarithm is continuous for all positive real numbers, we can simply substitute 0 into the expression to get \(\ln(7)\).

Step 4 ：For the third component, we have the limit of \(9t^2\) as \(t\) approaches 0. Since this is a polynomial function, we can also substitute 0 into the expression to get 0.

Step 5 ：Final Answer: The limit of the given vector as \(t\) approaches 0 is \(\boxed{1i + 1.9459101490553132j + 0k}\).