Question

$\underset{x\to a}{lim}\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{(3a+x)}-2\sqrt{\left(x\right)}}(a\ne 0).$ is $\frac{2}{b\sqrt{\left(c\right)}}$ Find value of $b\times c$

Answer 1

$\underset{x\to a}{lim}\frac{\sqrt{(a+2x)}-\sqrt{3x}}{\sqrt{(3a+x)}-2\sqrt{\left(x\right)}}$
$=\underset{x\to a}{lim}\frac{(a+2x)-3x}{(3a+x)-4x}\times \frac{\sqrt{(3a+x)}+2\sqrt{x}}{\sqrt{(a+2x)}+\sqrt{\left(3x\right)}}$
$=\underset{x\to a}{lim}\frac{a-x}{(a-x)}.\frac{\sqrt{(3a+2x)}+2\sqrt{\left(x\right)}}{\sqrt{(a+2x)}+\sqrt{\left(3x\right)}}$
$\underset{x\to a}{lim}\frac{1}{3}\frac{\sqrt{(3a+x)}+2\sqrt{\left(x\right)}}{\sqrt{(a+2x)}+\sqrt{\left(3x\right)}}=\frac{1}{3}.\frac{\sqrt{\left(4a\right)}+2\sqrt{\left(a\right)}}{\sqrt{\left(3a\right)}+\sqrt{\left(3a\right)}}$
$=\frac{1}{3}.\frac{4\sqrt{a}}{2\sqrt{\left(3\right)}\sqrt{\left(a\right)}}=\frac{2}{3\sqrt{\left(3\right)}}$
$\Rightarrow b\times c=3\times 3=9$