I don't believe this function's limit can be defined from the left as the greatest integer function jumps from when x is just less than 0 to when it is 0.
Had a look at the question again and you're correct Moo - I misread. It does exist as we only need to get arbitrarily close to x=0 to define the limit.
In which case we can consider the range where
for even so within this range
Also in this range we have that
and so since