Let f(t)=|(cost, t,1)(2sint,t,2t)(sint,t,t)|then find ​lim t→0 f(t)(t^2).

Question

Let \(f(t)=\begin{vmatrix} cos\,t & t & 1 \\[0.3em] 2sin\,t & t & 2t \\[0.3em] sin\,t &t & t \end{vmatrix}\), then find \(\lim\limits_{t \to0}\frac{f(t)}{t^2} \)

Answer 1

Given,

\(f(t)=\begin{vmatrix} cos\,t & t & 1 \\[0.3em] 2sin\,t & t & 2t \\[0.3em] sin\,t &t & t \end{vmatrix}\)

\(=\begin{vmatrix} cos\,t & t & 1 \\[0.3em] 0 & -t & 0 \\[0.3em] sin\,t &t & t \end{vmatrix}\) [Applying R₂ \(\longrightarrow\)R₂ - 2R₃]

\(=t\begin{vmatrix} cos\,t & 1 & 1 \\[0.3em] 0 & -1 & 0 \\[0.3em] sin\,t &1 & t \end{vmatrix}\)

Expanding along R₂, we get