For each x in R, let [x] be the greates...

For each ` x in R`, let [x] be the greatest integer less than or equal to x, Then,
` underset( x to 0^(-)) lim ( x ([x] + | x| ) sin [x]) / ([x])` is equal to

A

`sin 1`

B

`0`

C

`1`

D

`-sin 1`

Verified by Experts

The correct Answer is:

D

`lim_(xrarr0^(-))=(x([x]+|x|)sin[x])/(|x|)`
`lim_(xrarr0^(-))(-x(|x|-1)sin(1))/(|x|)`
`lim_(xrarr0)(-x(-x-1)sin(1))/(-x)=-sin1`

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