\[\mathop {\lim }\limits_{x \to 0} \frac{{1 + {x^2} - cosx}}{{{{\tan }^2}x}} = \mathop {\lim }\limits_{x \to 0} \frac{{2{{\sin }^2}\frac{x}{2} + {x^2}}}{{{{\tan }^2}x}}\]\[ = \mathop {\lim }\limits_{x \to 0} \frac{{2{{\sin }^2}\frac{x}{2} + {x^2}}}{{{x^2}}}.\frac{{{x^2}}}{{{{\tan }^2}x}} = \mathop {\lim }\limits_{x \to 0} \left[ {\frac{1}{2}{{\left( {\frac{{\sin \frac{x}{2}}}{{\frac{x}{2}}}} \right)}^2} + 1} \right].\frac{{{x^2}}}{{{{\tan }^2}x}} = \frac{3}{2}\]