Mathematics - Limits Question with Solution | TestHub

Given,

$\underset{x\to 3}{\lim }\frac{\sqrt{3x}-3}{\sqrt{2x-4}- \sqrt{2}}$

On rationalization we get,

$\underset{x\to 3}{\lim }\frac{\left(3x-9\right)\times \left(\sqrt{2x-4}+ \sqrt{2}\right)}{\left\{\left(2x-4\right)-2\right\}\times \left(\sqrt{3x}+3\right)}$

$= \underset{x\to 3}{\lim }\frac{3\left(x-3\right)}{2\left(x-3\right)}\times \frac{\sqrt{2x-4}+ \sqrt{2}}{\left(\sqrt{3x}+3\right)}$

$=\frac{3}{2}\times \frac{2\sqrt{2}}{6}=\frac{1}{\sqrt{2}}.$