There seems to be a lot of misunderstanding of limits in the answers here. Yes, this limit does exist.

Informally, in order for a limit of a function f(x) to exist at a point P, it has to have the same limit no matter how you get to P. But the key here is that you have to get there from within the domain of the function. For regular functions of one variable, there are at most two relevant directions then: from the left and from the right. Generally we need both of these limits to exist and agree. In this case though, x=0 is an endpoint of the function's domain. So, it's totally irrelevant what's going on for x<0. That has nothing to do with this function or limit at all. In such a case, we'd only have to consider the limit from the right.

A bit more formally, for the limit of f(x) not to exist at P there have to be two sequences of points {x_n},{y_n}, *taken from the domain of f*, such that

lim (n->∞) x_n = lim (n->∞) y_n = P

but

lim (n->∞) f(x_n) ≠ lim (n->∞) f(y_n).

All such sequences in this example have lim (n->∞) f(x_n) = 1, so we say indeed that lim (x->0) f(x) = 1.