Question

if
f(x)=x[x]/sin[x]
then show that lim f(x)has no existence
x>0

Answer 1

assuming you are using the [..] operator to have the same meaning as (..)
we have
Limit x^2/sinx , as x ---> 0
using L'Hopitals Rule
lim x^2/sinx
= lim 2x/cosx , as x --->
= 0/1
= 0

verification:
http://www.wolframalpha.com/input/?i=plot+y+%3D+x%5E2%2Fsinx

looks at x = 0,
the function has a value of 0 , the origin (0,0)

If [ ... ] is some other mathematical operator, define it please

Answer 2

as an example....

[2.32]=2