QJEEM19Mae6d5: JEE Main 2019 Mathematics - Class 12 JEEM19Mae6d5 Solution: For each x ∈ R , let [x] be the greatest integer less than or…

Question

Verified Solution • 2025–26 Session moderate

For each x

\in

R, let [x] be the greatest integer less than or equal to x.

Then

lim_{x \to 0^{-}} \frac{x ([x] + | x |) \sin [x]}{| x |}

is equal to :

Accepted Answer

lim_{x \to 0^{-}} \frac{x ([x] + | x |) \sin [x]}{| x |}

= lim_{h \to 0} \frac{(0 - h) ([0 - h] + | 0 - h |) \sin [0 - h]}{| 0 - h |}

= lim_{h \to 0} \frac{- h (- 1 + h) \sin (- 1)}{h}

= lim_{h \to 0} (- 1 + h) \sin (1) = - \sin 1

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