Let g(x) = lim{n→∞} (1+x/n)^n
g(0) = 1
dg(x)/dx = d/dx lim{n→∞} (1+x/n)^n = lim{n→∞} d/dx (1+x/n)^n
= lim{n→∞} n * (1+x/n)^(n-1) * (1/n) = lim{n→∞} (1+x/n)^n / (1+x/n)
= lim{n→∞} (1+x/n)^n = g(x)
Thus f(x) = g(x)
, especially f(1) = lim{n→∞} (1+1/n)^n