A question about the mathematical constant e.

I understand that when we write e = lim⁡ n→∞ (1+1/n)^n, we are identifying the number that this expression converges to as n becomes very large. However, I also know that lim⁡ n→∞(1/n)= lim⁡ n→∞(100/n )=0. Given this, why doesn't the same logic apply to e, such that lim⁡ n→∞ (1+1/n)^n = lim n→∞​ (1+100/n)^n = e?

I've seen the proof that shows:

Starting from the definition of e:

e = lim⁡ n→∞ (1+1/n)n

Exponentiating both sides by k:

e^k = (lim⁡ n→∞ (1+1/n)^n)^k

Using properties of limits and exponents:

e^k = lim⁡ n→∞ ((1+1/n)^n)^k

Simplifying the expression inside the limit:

((1+1/n)^n)^k = (1+1/n)^nk

Substituting m=nk:



Applying this to k=100:

e^100 = lim n→∞ ​(1+100/n)^n

But I want a more intuitive understanding of why lim⁡ n→∞ (1+1/n)^n = e while lim⁡ n→∞ (1+100/n)^n = e^100, even though 1/n and 100/n both approach zero as n tends to infinity.


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  • When you say "In fact 100/n is 100 times larger than 1/n, although both are very small numbers" Does that mean realistically 100/n and 1/n are different numbers, but the difference between them is infinitesimal allowing the difference to be ignored.

    • Martin Martin

      Yes, the difference is very small and can be ignored (before raising to the power of n of course).

  • Thanks so much for the explanation it has really helped me out.

    • Martin Martin

      You're welcome, happy it was helpful.

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