Help in LIMITS

i got the answr as 1....bt the answer is given 0 in the ARIHANT DIFFERENTIAL CALCULUS book

Try using the expansion of sinx and you will notice that sinx/x will always be less than 1 which impiles that greatest integer satisfying this is 0.

or simply,

consider f(x) = sinx - x
f(0) = 0
f'(x) = cosx - 1 < 0 for all x

=> sinx - x is decreasing for x > 0

=> sinx < x for x > 0
=> |sinx| < |x| for all x ≠0

=> |sinx|/|x| < 1 for all x≠0

hence [|sinx||x|] = 0 for all x ≠0

and since you are calculating limit, lim(x→0) ≠x = 0

there fore x is non-zero hence the limit = 0

but if we calculate the limit using the conventional methods,

both the L.H.L.and the R.H.L comes to 0

why the difference?

the correct ans. is 0.