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One of the standard definitions of $e$ is as

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

But in all cases I’ve seen this limit, it is proven as a limit of the sequence $\Big\{\big(1 + \frac{1}{n}\big)^n\Big\}$, which seems to cover the limit for only $n$ as an integer. Now my question is whether this sequence limit is equivalent to a normal limit for which $n$ can be any real number. I can think of cases which this isn’t generally true;

- Evaluate $\int \sqrt{ \frac {\sin(x-\alpha)} {\sin(x+\alpha)} }\,\operatorname d\!x$?
- Evaluating $\displaystyle\int_0^1\frac{\sqrt{1-y^2}}{1+y^2}dy$ without trig substitution
- Integration of $\displaystyle \int\frac{1}{1+x^8}\,dx$
- If a function has a finite limit at infinity, does that imply its derivative goes to zero?
- Integrate $I=\int_0^1\frac{\arcsin{(x)}\arcsin{(x\sqrt\frac{1}{2})}}{\sqrt{2-x^2}}dx$
- Behaviour of ($\overline{a}_n)_{n=1}^{\infty}$ and ($\underline{a}_n)_{n=1}^{\infty}$

$$\lim_{n\rightarrow\infty}\ \sin(n\pi)$$

comes readily to mind, for which the limit as a sequence is simply $0$ but as a general limit, it is undefined. The limit for $e$ is used exactly as if it were a normal limit, which leads me to believe it is equivalent. Are there conditions for which the limit of a sequence and the corresponding function are identical?

- Evaluate $\int_0^1x\log\left(1+x^2\right)\left^3\operatorname{d}\!x$
- Calculate a limit $\lim_{x \to \pi/2} \frac{\sqrt{ \sin x} - \sqrt{ \sin x}}{\cos^2x}$
- Prove: If a sequence converges, then every subsequence converges to the same limit.
- Improper integral with log and absolute value: $\int^{\infty}_{0} \frac{\log |\tan x|}{1+x^{2}} \, dx$
- Divergence of $\sum\limits_n1/\max(a_n,b_n)$
- lim$_{n\rightarrow \infty}\int _{-\pi}^\pi f(t)\cos nt\,dt$
- If a linear operator has an adjoint operator, it is bounded
- Uniform convergence and integration
- Compute $\int_0^\pi\frac{\cos nx}{a^2-2ab\cos x+b^2}\, dx$
- Exists $C = C(\epsilon, p)$ where $\|u\|_{L^\infty(0, 1)} \le \epsilon\|u'\|_{L^p(0, 1)} + C\|u\|_{L^1(0, 1)}$ for all $u \in W^{1, p}(0, 1)$?

If $x>0$ be a real number, then write $x=n+y_x$ where $0 \leq y_x \leq 1$.

It is easy then to show that

$$ \left(1+\frac{1}{n+1} \right)^n \leq \left(1+\frac{1}{x}\right)^x \leq \left(1+\frac{1}{n}\right)^{n+1} \,.$$

Using this, you can prove that if

$$e=\lim_n \left(1+\frac{1}{n}\right)^n$$

then

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

**Edit**

To complete the answer, in general if $f(x)$ is monotonic, then for sure

$$\lim_n f(n)= \lim_x f(x) \,.$$

I don’t recall if $\left(1+\frac{1}{x}\right)^x$ is monotonic (and I am too lazy to differentiate it), but what we showed above is that it is at least “close to being monotonic”. What I mean by this, I showed that $$f(n) h(n) \leq f(x) \leq f(n+1)g(n+1) \forall x\in [n, n+1)$$ where $h(n)$ and $g(n)$ go to $1$. The you basically sqeeze it.

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- $A\otimes_{\mathbb C}B$ is finitely generated as a $\mathbb C$-algebra. Does this imply that $A$ and $B$ are finitely generated?