Discussion
Value of \(\sqrt{3.\sqrt{3.\sqrt{3.\sqrt{3....}}}}\)
Answer: A
The value is 3.
Let the infinite expression equal \(x\). Because it repeats without end, the part under the first root sign is again \(x\):
\[ \begin{aligned} x &= \sqrt{3x} \\ x^{2} &= 3x \Rightarrow x = 3 \end{aligned} \]
(\(x = 0\) is rejected because the expression is clearly positive.) Another way: the expression is \(3^{\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots} = 3^{1} = 3\). It does not grow to infinity, and \(\sqrt{3}\) is only the value of the innermost root.
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