If you allow complex values, then it is alright to use a two-sided limit but then if you are allowing such things, you might as well consider the limit on the complex plane. This might be a bit of a tangent but here we go. Let
for complex values . We'll show that is analytic. The condition with the Cauchy-Riemann equations is equivalent to saying that
for
If you calculate this, it is indeed equal to zero. So the function is analytic and we can take the limit as . The limit happens to be real and unbounded in the positive direction.