The value of \mathop {\lim }\limits_{x o {0^ + }} {{{{\cos }^{ - 1}}(x - {{[x]}^2}).{{\sin }^{ - 1}}(x - {{[x]}^2})} \over {x - {x^3}}}, where [ x ] denotes the greatest integer \le x is :

Question

The value of \mathop {\lim }\limits_{x 	o {0^ + }} {{{{\cos }^{ - 1}}(x - {{[x]}^2}).{{\sin }^{ - 1}}(x - {{[x]}^2})} \over {x - {x^3}}}, where [ x ] denotes the greatest integer \le x is :

Accepted Answer

Answer: C. {\pi \over 2} \mathop {\lim }\limits_{x 	o {0^ + }} {{{{\cos }^{ - 1}}\left( {x - {{[x]}^2}} ight).{{\sin }^{ - 1}}\left( {x - {{[x]}^2}} ight)} \over {x - {x^3}}} = \mathop {\lim }\limits_{x 	o {0^ + }} {{{{\cos }^{ - 1}}x} \over {1 - {x^2}}}.{{{{\sin }^{ - 1}}x} \over x} = {\cos ^{ - 1}}0 = {\pi \over 2}

The value of $$\mathop {\lim }\limits_{x \to {0^ + }} {{{{\cos }^{ - 1}}(x - {{[x]}^2}).{{\sin }^{ - 1}}(x - {{[x]}^2})} \over {x - {x^3}}}$$, where [ x ] denotes the greatest integer $$ \le $$ x is :

Answer

Explanation

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