JEE Main 2017 (Online) 8th AprilQuestion 1
$\lim_{x\to3}\frac{\sqrt{3x}-3}{\sqrt{2x-4}-\sqrt2}$ is equal to :
Limits & Continuity
Solution
Concept. At $x=3$: $\sqrt9-3=0$ and $\sqrt2-\sqrt2=0$; use L'Hospital. Formulas used. (1) $\frac{d}{dx}\sqrt{px+q}=\frac{p}{2\sqrt{px+q}}$(2) L'Hospital for $\frac00$Step 1 - Numerator derivative (1). $\frac{3}{2\sqrt{3x}}$; at $3$: $\frac{3}{6}=\frac12$. Step 2 - Denominator derivative. $\frac{2}{2\sqrt{2x-4}}$; at $3$: $\frac{1}{\sqrt2}$. Step 3 - Ratio (2). $\frac{1/2}{1/\sqrt2}=\frac{\sqrt2}{2}=\frac{1}{\sqrt2}$. Answer: A