The expression \(\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+.......}}}}}\) extends to an infinite number of roots. Which of the following choices most closely approximates the value of this expression?
Explanation: Let \(x = \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+.......}}}}}\). Then \(x = \sqrt{2+x}\). \(x^{2} = 2+x \Rightarrow x^{2}-x-2=0 \Rightarrow (x-2)(x+1)=0\). Since \(x\) cannot be negative, \(x = 2\).
