If you are doing real analysis, this proviso means that you don't run into any problems - the exponential function can be defined.
As others have noted in threads linked in the comments, it is possible to extend the definition of the exponential function, but there are some technicalities involved which impinge on whether it makes sense to take a limit at all - whether the function you define is continuous, for example.
And don't forget that in taking the limit you have specified $g(x)$ varies as well as $f(x)$ so you are potentially dealing with a continuously changing real exponent rather than some nice integer or rational one - so you really need the exponential function to be valid for all reals, and continuous in the exponent. If $f(x)$ is positive, there isn't a problem.
This doesn't mean it can't be done for other values, of course. But there are pitfalls for the unwary.