Question

If $f(x)=\{\begin{array}{ll}\frac{\sin 2x}{cx}+\frac{x}{4\left(\sqrt{x+a^2}-a\right)}& ;x\ne 0,(a<0,c\ne 0)\\ b& ;x=0,(b\ne 0)\end{array}$ andand $f\left(x\right)$ is continuous at $x=0$ , then the value of $bc$ is equal to

Answer 1

Answer: 2

$\because f\left(x\right)$ is continuous at $x=0$
$\therefore \underset{x\to 0}{\lim }f\left(x\right)=f\left(0\right)$
$\underset{x\to 0}{\lim }\frac{sin2x}{cx}+\frac{x}{4\left(\sqrt{x+a^2}-a\right)}\left(\frac{\sqrt{x+a^2}+a}{\sqrt{x+a^2}+a}\right)=b$
$\underset{x\to 0}{\lim }\frac{2}{c}+\frac{x\left(\sqrt{x+a^2}+a\right)}{4\left(\left(x+a^2\right)-a^2\right)}=b\Rightarrow \frac{2}{c}+0=b(asa<0)$
$\Rightarrow bc=2$