Step 1: Compare the growth rates.
Inside the nested root, the dominant term for large $x$ is the outer $x$; the inner roots grow much slower. So the denominator behaves like $\sqrt{x}$.
Step 2: Factor $x$ out of the denominator.
Write the denominator as $\sqrt{x}\sqrt{1+\frac{\sqrt{x+\sqrt x}}{x}}$.
Step 3: Cancel the leading $\sqrt{x}$.
The numerator $\sqrt{x}$ cancels, leaving $\frac{1}{\sqrt{1+\frac{\sqrt{x+\sqrt x}}{x}}}$.
Step 4: Estimate the inner fraction.
The numerator $\sqrt{x+\sqrt x}\approx\sqrt{x}$ for large $x$, so $\frac{\sqrt{x+\sqrt x}}{x}\approx\frac{\sqrt x}{x}=\frac{1}{\sqrt x}$.
Step 5: Take the limit of the inner part.
As $x\to\infty$, $\frac{1}{\sqrt x}\to 0$, so the term under the inner root tends to $0$.
Step 6: Evaluate the whole expression.
The limit becomes $\frac{1}{\sqrt{1+0}}=1$, option 3.
\[ \boxed{1} \]