For some a, b, let f(x)=|arrayccc a+( x)/(x) & 1 & ~b a & 1+( x)/(x) & ~b a & 1 & ~b+( x)/…

Question

For some $a, b,$ let $f(x)=\left|\begin{array}{ccc} \mathrm{a}+\frac{\sin x}{x} & 1 & \mathrm{~b} \ \mathrm{a} & 1+\frac{\sin x}{x} & \mathrm{~b} \ \mathrm{a} & 1 & \mathrm{~b}+\frac{\sin x}{x} \end{array}ight|, x 
eq 0, \lim \limits_{x ightarrow 0} f(x)=\lambda+\mu \mathrm{a}+
u \mathrm{b}.$ Then $(\lambda+\mu+v)^2$ is equal to :

Accepted Answer

$$\begin{aligned}
& \lim _{x \rightarrow 0} f(x)=\left|\[\begin{array}{ccc} a+1 & 1 & b \ a & 1+1 & b \ a & 1 & b+1 \end{array}\end{aligned}$$\right| \
\& =(a+1)(2(b+1)-b)+1(a b-a(b+1))+b a \
\& =(a+1)(b+2)-a+a b \
\& =b+a+2=\lambda+\mu a+v b \
\& \lambda=2, \mu=1, v=1 \Rightarrow(\lambda+\mu+v)\^2=16
\]

Solutions · 1