Question

Evaluate: \[ \lim_{x \to 0} \frac{\sqrt{11 + |x|} - 6\sqrt{2 + |x|}}{6 - 2\sqrt{2 + |x|}} \]

• A. \( -1 \)
• B. \( -\frac{1}{2} \)
• C. \( \frac{\sqrt{11 - 6\sqrt{2}}}{3 - \sqrt{2}} \)
• D. \( \frac{1}{2} \)

Hint:

Rationalizing Roots}
Always simplify the limit before substituting
Use rationalization to simplify expressions involving roots
Don’t forget \( |x| = x \) when \( x \to 0^+ \) or \( -x \) when \( x \to 0^- \)

Answer 1

The Correct Option is D

Let \( x \to 0 \Rightarrow |x| \to 0 \) So: \[ \lim_{x \to 0} \frac{\sqrt{11 + 0} - 6\sqrt{2 + 0}}{6 - 2\sqrt{2 + 0}} = \frac{\sqrt{11} - 6\sqrt{2}}{6 - 2\sqrt{2}} \] Rationalizing the denominator: \[ = \frac{\sqrt{11} - 6\sqrt{2}}{6 - 2\sqrt{2}} \cdot \frac{6 + 2\sqrt{2}}{6 + 2\sqrt{2}} = \cdots = \frac{1}{2} \]