Question

Let $f(t)=\left|\begin{array}{ccc}\cos t& t& 1\\ 2\sin t& t& 2t\\ \sin t& t& t\end{array}\right|$, then $\underset{t\to 0}{\lim }\frac{f(t)}{t^2}$ equals to

• A. 0
• B. -1
• C. 2
• D. 3

Answer 1

Answer: (A)

We have $f(t)=\left|\begin{array}{ccc}\cos t& t& 1\\ 2\sin t& t& 2t\\ \sin t& t& t\end{array}\right|$,
Expanding $c_1$, we can get
$=-t^2\cos t+t\sin t$
Now, $\underset{t\to 0}{\lim }\frac{f(t)}{t^2}=\underset{t\to 0}{\lim }\frac{-t^2\cos t}{t^2}+\underset{t\to 0}{\lim }\frac{t\sin t}{t^2}$
$=-1+1=0$