$\lim _{x \rightarrow \frac{1}{\sqrt{2}}} \frac{\sin \left(\cos ^{-1} x\right)-x}{1-\tan \left(\cos ^{-1} x\right)}$ is equal to :

JEE Main · 2022 · 26 Jun 2022 · Shift 1 · Limits

lim_{x \to \frac{1}{2}} \frac{s i n ( c o s ^{- 1} x ) - x}{1 - t a n ( c o s ^{- 1} x )}

is equal to :

Answer: (D)

Let

θ = cos^{- 1} x

, so

x = cos θ

and

sin θ = 1 - x^{2}

.

The expression becomes:

lim_{x \to \frac{1}{2}} \frac{1 - x ^{2} - x}{1 - \frac{1 - x ^{2}}{x}} = lim_{x \to \frac{1}{2}} \frac{x ( 1 - x ^{2} - x )}{x - 1 - x ^{2}} = lim_{x \to \frac{1}{2}} (- x) = - \frac{1}{2}

JEE Main · 2022 · 26 Jun 2022 · Shift 1 · Single correct option

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