If does not exist, then sum of the all possible values of k is[Note : [y] denotes greatest integer function less than or equal to y.]

Engineering

Mathematics

Trigonometric Limits

Question

If $\underset{x \to k}{Lim} [\sin^{- 1} x + \cos^{- 1} x + \tan^{- 1} x]$ does not exist, then sum of the all possible values of k is

[Note : [y] denotes greatest integer function less than or equal to y.]

$\frac{- \sin 1}{\sin 2 \cdot \sin 3}$

$\frac{- \sin 2}{\sin 1 \cdot \sin 3}$

$\frac{- \sin 3}{\sin 1 \cdot \sin 2}$

$\frac{- \sin 1 \cdot \sin 2}{\sin 3}$

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Range of sin^–1x + cos^–1x + tan^–1x

$= \frac{π}{2} + \tan^{- 1} \forall x \in [- 1, 1]$

$∴ R_{f} = [\frac{π}{4}, \frac{3 π}{4}]$

Range contains 2 integers 1, 2.

$∴ \frac{π}{2} + \tan^{- 1} x = 1 \Rightarrow x = - \cot 1$

and $\frac{π}{2}$ + tan^-1x = 2 ⇒ x = – cot 2

Sum = – (cot 1 + cot 2) = $(\frac{\cos 1}{\sin 1} + \frac{\cos 2}{\sin 2}) = - (\frac{\sin 3}{\sin 1 \cdot \sin 2})$