MCQ
$\mathop {\lim }\limits_{x \to 1} \frac{{1 - \sqrt x }}{{{{({{\cos }^{ - 1}}x)}^2}}} = $
- A$1$
- B$\frac{1}{2}$
- ✓$\frac{1}{4}$
- DNone of these
$\mathop {\lim }\limits_{x \to 1} \,\frac{{1 - \sqrt x }}{{{{({{\cos }^{ - 1}}x)}^2}}} = \mathop {\lim }\limits_{y \to 0} \,\frac{{1 - \sqrt {\cos y} }}{{{y^2}}}$
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} }} $
$= \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}.$
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