A Taylor approximation needs an error estimate to say how accurate it is.
Every derivative of eˣ equals eˣ. On the interval between 0 and x, their magnitude is at most e^{max(0,x)}. Taylor’s theorem therefore bounds the degree-n Maclaurin remainder by that maximum times |x|^{n+1}/(n+1)!. The actual error is computed separately.
Using a derivative bound from the wrong interval can invalidate the estimate.