f[x_] = (Exp[-(1 + x)^2] - Exp[-1^2])/x

10 (1/E^(121/100) - 1/E)

Table[1/(10^i), {i, 14}]

f[0.1]
(* -0.696822 *)

f[x_] = (Exp[-(1. + x)^2] - Exp[-1^2])/x
f[1/10]
(* -0.696822 *)

f[x_] = N[(Exp[-(1 + x)^2] - Exp[-1^2])/x]
f[1/10]
(* -0.696822 *)