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Q.
Let $f(t)=\begin{vmatrix}\cos t & t & 1 \\ 2 \sin t & t & 2 t \\ \sin t & t & t\end{vmatrix}$, then $\displaystyle\lim _{t \rightarrow 0} \frac{f(t)}{t^{2}}$ equals to
Solution:
We have $f(t)=\begin{vmatrix}\cos t & t & 1 \\ 2 \sin t & t & 2 t \\ \sin t & t & t\end{vmatrix}$,
Expanding $c_{1}$, we can get
$=-t^{2} \cos t+t \sin t$
Now, $\displaystyle\lim _{t \rightarrow 0} \frac{f(t)}{t^{2}}=\displaystyle\lim _{t \rightarrow 0} \frac{-t^{2} \cos t}{t^{2}}+\displaystyle\lim _{t \rightarrow 0} \frac{t \sin t}{t^{2}}$
$=-1+1=0$