Sx Sin[1/x] is caught between x and -x. By the Flyswatter principle,
the limit as x--> is 0.
The function blows up and occilates infinitely often. No limit.
As x--> 0, the limits along irrations x and along rational x are the same,p namely zero
Limit is therefore zero.
At x = &pgr; the limits are different
Left limit is 4, Right limit if 3
Limit does not exist (Infinity is not a real number)
This can be seen as = ( Cos[x] -1)