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 odd
for even so within this range
Also in this range we have that
and so since
when