Evaluation recursive limit

Question

How to evaluate this limit:

$\lim_{x\to0^+}\dfrac { -1+\sqrt { \tan(x)-\sin(x)+\sqrt { \tan(x)-\sin(x)+\sqrt { \tan(x)-\sin(x) } +...\infty } } }{ -1+\sqrt { { x }^{ 3 }+\sqrt { { x }^{ 3 }+\sqrt { x^{ 3 } } +...\infty } } } $

Answer 1

HINT:

Let $\sqrt { \tan(x)-\sin(x)+\sqrt { \tan(x)-\sin(x)+\sqrt { \tan(x)-\sin(x) } +...\infty } }=y$

$\implies\tan(x)-\sin(x)+y=y^2$

$\implies y=?$

Similarly, for $\sqrt { { x }^{ 3 }+\sqrt { { x }^{ 3 }+\sqrt { x^{ 3 } } +...\infty } } $