lim x → 1 2 sin cos - 1 x - x 1 - tan cos - 1 x is equal to, 1 2, - 1 2, 2, - 1

Let cos − 1 x = t i.e. x = cos t So lim t → π 4 sin t − cos t 1 − tan t = lim t → π 4 − cos t = − 1 2

lim x → 1 2 sin cos - 1 x - x 1 - tan cos

Question: Answered & Verified by Expert

\lim_{x \to \frac{1}{\sqrt{2}}} \frac{\sin (\cos^{- 1} x) - x}{1 - \tan (\cos^{- 1} x)}

is equal to

MathematicsLimitsJEE MainJEE Main 2022 (26 Jun Shift 1)

Options:

Solution:

1627 Upvotes Verified Answer

The correct answer is:

\frac{- 1}{\sqrt{2}}

Let $\cos^{- 1} x = t$ i.e. $x = \cos t$

So $\lim_{t \to \frac{π}{4}} \frac{\sin t - \cos t}{1 - \tan t}$ $= \lim_{t \to \frac{π}{4}} (- \cos t) = \frac{- 1}{\sqrt{2}}$

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