I'm not 100% sure what the "

" method is, but it's trivial to calculate the limit as x approaches zero
from the positive side. At some very, very small positive number,

, the function value is 1. No matter how small

becomes, as long as it's positive, the function value is always 1. Hence the limit from the positive side is 1.
Meanwhile, it's also trivial to calculate the limit
from the negative side. Using the same arguments as above, but with some small number

, it's easy to see that the limit from the negative side is -1.
Since the limit is different, depending whether you approach zero from the positive side or from the negative side, the actual limit is undefined.