The infinite series is not an approximation and it is valid for all values of x, both real and complex. Don't factor out the x and set the initial condition that e^0 = 1 which gives c = 1 which is the first term x^0/0! and we the original series back -term by term. This is not a trick and integrating term by term in a convergent infinite series is valid process.

Again, e^x is actually equal to x^0/0! + x^1/1! + x^2/2! + ... to an infinite number of terms and is not an approximation.

Not everything can be verified by this trick, as we all know that the series is just approximation.