For some a,b,R, let [f(x)=vmatrixa+ xx & 1 & b a & 1+ xx & b a & 1 & b+ xxvmatrix, x0, _x0…

Question

For some $a,b,\nu\in\mathbb{R}$, let
$$f(x)=\begin{vmatrix}a+\dfrac{\sin x}{x} & 1 & b\ a & 1+\dfrac{\sin x}{x} & b\ a & 1 & b+\dfrac{\sin x}{x}\end{vmatrix},\ x\ne0,\qquad \lim_{x\to0}f(x)=\lambda+\mu a+\nu b.$$
Then $(\lambda+\mu+\nu)^2$ is equal to:

Accepted Answer

As $x\to0$, $\dfrac{\sin x}{x}\to1$. Expanding the determinant using row/column reduction and taking the limit gives
$$\lim_{x\to0}f(x)=(a+1)(b+2)-a+ab = a+b+2.$$

So $\lambda=2,\ \mu=1,\ \nu=1$, giving $(\lambda+\mu+\nu)^2=4^2=16$.

Solutions · 1