$\underline{\textbf{Solution 1}}$ Let $S=\sqrt{x\sqrt{x\sqrt{x\sqrt{\cdots}}}}$, then squaring both sides leads $$S^2 = x\sqrt{x\sqrt{x\sqrt{x\sqrt{\cdots}}}}\implies S^2 = xS \implies S=x$$ $\underline{\textbf{Solution 2}}$ Rewriting the left side of the given expression as exponential form gives $$\sqrt{x\sqrt{x\sqrt{x\sqrt{\cdots}}}}=x^{1/2}x^{1/4}x^{1/8}\cdots=x^{1/2+1/4+1/8+\cdots}= x^1$$