Find the limit.
$\lim_{x\rarr3}\frac{\sqrt{x}-\sqrt3}{x-3}$

• A. $2\sqrt3$
• B. $0$
• C. $\frac{\sqrt3}{6}$
• D. Does not exist

Answer: (C)

Verified step by step guidance

1. Identify the form of the limit: As x approaches 3, both the numerator and the denominator approach 0, indicating an indeterminate form 0/0. This suggests that we need to simplify the expression.
2. Rationalize the numerator: Multiply the numerator and the denominator by the conjugate of the numerator, which is (\(\sqrt{x} + \sqrt{3}\)). This will help eliminate the square roots in the numerator.
3. Simplify the expression: After multiplying, the numerator becomes \((x - 3)\) because \((\sqrt{x} - \sqrt{3})(\sqrt{x} + \sqrt{3}) = x - 3\). The denominator becomes \((x - 3)(\sqrt{x} + \sqrt{3})\).
4. Cancel the common factor: The \((x - 3)\) terms in the numerator and denominator cancel each other out, leaving \(\frac{1}{\sqrt{x} + \sqrt{3}}\).
5. Evaluate the limit: Substitute \(x = 3\) into the simplified expression \(\frac{1}{\sqrt{x} + \sqrt{3}}\) to find the limit as x approaches 3.