\displaystyle\lim_{x\to\infty} \dfrac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}}=
y=\sqrt {x}olsun, x\rightarrow \infty \Rightarrow y\rightarrow \infty
\lim _{y\rightarrow \infty }\dfrac {y}{\sqrt {y^{2}+\sqrt {y^{2}+y}}}=\lim _{y\rightarrow \infty }\dfrac {y}{\sqrt {y^{2}+y\sqrt {1+\dfrac {1}{y}}}} =
\lim _{y\rightarrow \infty }\dfrac {y}{y\sqrt {1+\dfrac {\sqrt {1+\dfrac {1}{y}}}{y}}}
= 1