For each x ε R, let [x] be the greatest integer less than or equal to x. Then displaystyle limx→0- (x([x] +[x]) sin[x]/|x|) is equal to

Question

For each x ε R, let [x] be the greatest integer less than or equal to x. Then displaystyle limx→0- (x([x] +[x]) sin[x]/|x|) is equal to (A)  - sin 1 (B) 0 (C) 1 (D)  sin 1

Tardigrade Admin · Accepted Answer

limx→0- (x([x]+|x|) sin[x]/|x|) x →0- [x] = - 1 ⇒ limx→0- (x(-x-1) sin(-1)/-x) = - sin 1 |x | = - x

Q. For each $x εR$ , let $[x]$ be the greatest integer less than or equal to $x$ . Then $x \to 0^{-} lim \frac{x ( [ x ] + [ x ] ) sin [ x ]}{∣ x ∣}$ is equal to

$- sin 1$

$0$

$1$

$sin 1$

$lim_{x \to 0^{-}} \frac{x ( [ x ] + ∣ x ∣ ) s i n [ x ]}{∣ x ∣}$
$x \to 0^{-}$
$[x] = - 1 \Rightarrow lim_{x \to 0^{-}} \frac{x ( - x - 1 ) s i n ( - 1 )}{- x} = - sin 1$
$∣ x ∣ = - x$

Q. For each xεR, let [x] be the greatest integer less than or equal to x. Then x→0−lim​∣x∣x([x]+[x])sin[x]​ is equal to

−sin1

0

1

sin1

limx→0−​∣x∣x([x]+∣x∣)sin[x]​ x→0− [x]=−1⇒limx→0−​−xx(−x−1)sin(−1)​=−sin1 ∣x∣=−x

Q. For each $x εR$ , let $[x]$ be the greatest integer less than or equal to $x$ . Then $x \to 0^{-} lim \frac{x ( [ x ] + [ x ] ) sin [ x ]}{∣ x ∣}$ is equal to

$- sin 1$

$0$

$1$

$sin 1$

$lim_{x \to 0^{-}} \frac{x ( [ x ] + ∣ x ∣ ) s i n [ x ]}{∣ x ∣}$
$x \to 0^{-}$
$[x] = - 1 \Rightarrow lim_{x \to 0^{-}} \frac{x ( - x - 1 ) s i n ( - 1 )}{- x} = - sin 1$
$∣ x ∣ = - x$