Why does $\lim_{n\to\infty}(1+\frac{1}{n})^n = e$ instead of $1$?

Question

On Wikipedia, it says that $\lim_{n\to\infty}(1+\frac{1}{n})^n = e$ :

It [e] is approximately equal to 2.71828,[1] and is the limit of (1 + 1/n)n as n approaches infinity, ... (Source)

When I evaluate $(1+\frac{1}{n})^n$ for $n = 10^8$, I get approximately $2.718281798347$ which indeed is pretty close to $e$.

But when I try to "solve" the limit using the laws of limits, I get

$$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = \left(\lim_{n\to\infty}\left(1+\frac{1}{n}\right)\right)^n$$

because of the power law

$$ = \left(\lim_{n\to\infty}(1)+\lim_{n\to\infty}\left(\frac{1}{n}\right)\right)^n$$

because of the addition law

$$=\left(1+0\right)^n = 1$$

but that would mean that $e=1$, which is obviously not true.

What am I missing / doing wrong?

Thanks in advance.

Answer 1

$$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = \left(\lim_{n\to\infty}\left(1+\frac{1}{n}\right)\right)^n$$

This is wrong. The quantity on the left is independent of $n$ while the one on the right is not.

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What the power law tells you is for some positive integer $m$, $$ \lim_{n\to\infty}a_n^m=(\lim_{n\to\infty}a_n)^m $$ if $a_n>0$ for all $n$ and $\lim_{n\to\infty}a_n$ exists.

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[Added later:] Despite the incorrect reasoning mentioned above, it is worth to note that the limit $\lim_{n\to\infty}(1+1/n)^n$ is sometimes used as a definition of $e$. Therefore, unless you know why the limit is not $1$, it would be "unfair" to say that "e=1" is "obviously not true".

Also, you originally put the Euler-constant tag to your question, which is also incorrect. Since the constant $e$, sometimes called the "Euler number" is not the same as the Euler's constant.

Answer 2

You can see that the limit is greater than or equal to two, simply by using the first two terms of the binomial expansion, and noting that the omitted terms are positive. i.e. for each $n\gt 1$ separately: $$\left(1+\frac 1n\right)^n=1+n\cdot\frac 1n+ \text {other terms}\gt 1+1$$

The source of the error in reasoning is expressed by others, but this simple check tells you that you are wrong.

Answer 3

The so-called "power law" that you refer to, e.g. something like

If $n$ is an integer and $\lim\limits_{x\to a} f(x)$ exists, then $$\lim\limits_{x\to a}(f(x))^n = (\lim \limits_{x\to a} f(x))^n.$$

It refers specifically to a constant $n$ that does not change with the limit. Instead you have something like $$\lim\limits_{n\to\infty}(f(n))^n $$ where both the power and the inner function depends on $n$. In the specific example of $e$ we have that $$f(n) = 1+\frac1n.$$