Question

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

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

Hint:

Whenever an expression contains nested square roots, try substituting the inner square root by a variable. It often converts the expression into a perfect square.

Answer 1

The Correct Option is D

Step 1: Substitute t = sqrt(2+|x|).
As \(x\to0\), \(t\to\sqrt{2}\). Numerator becomes \(\sqrt{(t-3)^2}=3-t\) (since \(t<3\)). Denominator \(=6-2t=2(3-t)\).

Step 2: Cancel and evaluate.
\[\frac{3-t}{2(3-t)}=\frac{1}{2}\to\boxed{\dfrac{1}{2}}\]