When n tends to infinity, (1+1/n)^n tends to an irrational number, and this number does not appear in elementary mathematics, so it is defined as e, and e is approximately equal to 2.71828, which is an infinite non-repeating decimal and a transcendental number.
lim n→0, (1 + 1/n)^n.
=e^lim n→0, nln(1+1/n).
=e^lim n→0,1/n*ln(1+1/n).
=(Luo)e^lim n→0,1/1+1/n.
=e^0.
=1.
Standard definition of the limit of a sequence:
For the sequence {xn}, if there is a constant a, for any ε>0, there always exists a positive integer N, so that when n>N, | xn-a|<ε holds, then a is said to be the limit of the sequence {xn}.
Standard definition of function limit: Suppose function f(x), |x| is defined when it is greater than a certain positive number. If there is a constant A, for any ε>0, there will always be a positive integer X, such that when When x>X, |f(x)-A|<ε holds, then A is said to be the limit of function f(x) at infinity.
Suppose the function f(x) is defined in a certain decentered neighborhood of x0. If there is a constant A, for any ε>0, there will always be a positive number δ, such that when |x-xo| <δ, |f(x)-A|<ε holds, then A is said to be the limit of function f(x) at x0.