$\mathop {\lim }\limits_{x \to 0} \,\frac{{\tan \,(\pi \,{{\sin }^2}\,x) + \,{{(\left| x \right|\, - \,\sin \,(x\,[x]))}^2}}}{{{x^2}}}$
$1\left( \pi \right).{\left( 1 \right)^2} + \mathop {Lt}\limits_{x \to 0} {\left( {1 - \frac{{\sin x\left[ x \right]}}{{x\left[ x \right]}}.\frac{{x\left[ x \right]}}{{\left| x \right|}}} \right)^2}\,\,\,\,\,\,\,\,......\left( i \right)$
$\mathop {Lt}\limits_{x \to {0^ - }} \frac{{x\left[ x \right]}}{{\left| x \right|}} = \frac{x}{{ - x}}\left( { - 1} \right) = 1$
$\mathop {Lt}\limits_{x \to {0^ + }} \frac{{x\left[ x \right]}}{{\left| x \right|}} = \frac{{x\left( 0 \right)}}{x} = 0$
Put in equation $(i)$
$\therefore $ Limit does not exist.
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