Question

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

Answer 1

$\because f\left(x\right)$ is continuous at $x=0$
$\therefore \underset{x \rightarrow 0}{l i m}f\left(x\right)=f\left(0\right)$
$\underset{x \rightarrow 0}{l i m}\frac{sin 2 x}{c x}+\frac{x}{4 \left(\sqrt{x + a^{2}} - a\right)}\left(\frac{\sqrt{x + a^{2}} + a}{\sqrt{x + a^{2}} + a}\right)=b$
$\underset{x \rightarrow 0}{l i m}\frac{2}{c}+\frac{x \left(\sqrt{x + a^{2}} + a\right)}{4 \left(\left(x + a^{2}\right) - a^{2}\right)}=b\Rightarrow \frac{2}{c}+0=b\left(\right.asa < 0\left.\right)$
$\Rightarrow bc=2$