STD 11 - 12. limits — Maths STD 11 Science — Question

$\mathop {\lim }\limits_{x \to 1} \,\frac{{1 - \sqrt x }}{{{{({{\cos }^{ - 1}}x)}^2}}} = \mathop {\lim }\limits_{y \to 0} \,\frac{{1 - \sqrt {\cos y} }}{{{y^2}}}$

Now rationalizing it, we get

$\mathop {\lim }\limits_{y \to 0} \,\frac{{(1 - \cos y)}}{{{y^2}(1 + \sqrt {\cos y} )}}$

$ = \mathop {\lim }\limits_{y \to 0} \,\frac{{1 - \cos y}}{{{y^2}}}\,.\,\mathop {\lim }\limits_{y \to 0} \,\frac{1}{{1 + \sqrt {\cos y} }} $

$= \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}.$