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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 } } } $

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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 } } $

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  • $\begingroup$ So its evaluating to 1/2 right? Thanks a ton :-) $\endgroup$
    – user220382
    Jun 22, 2015 at 12:30

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