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We know that $\lim_\limits{n \to \infty} \left(1+ \frac{x}{n}\right)^n = e^x$, and this implies that $e = \lim_\limits{n \to \infty} \left(1+ \frac{1}{n}\right)^{n} $

However, if you raise both sides of that latter equation to the power of $x$, we get $$e^x = \lim_\limits{n \to \infty} \left(1+ \frac{1}{n}\right)^{nx} $$

This implies that: $$\lim_\limits{n \to \infty} \left(1+ \frac{x}{n}\right)^n =\lim_\limits{n \to \infty} \left(1+ \frac{1}{n}\right)^{nx}$$

Without using the fact that these limits both converge to $e^x$, how can you prove that they are equal to each other? Is there a way to algebraically manipulate one side so that it equals the other side?

Mailbox
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  • It's not true that $\left(1+\frac{x}{n}\right)^n = \left(1+\frac{1}{n}\right)^{nx}$, if that's what you mean. That doesn't rule out another method with algebraic manipulation leading to a simpler limit, but I don't see one immediately. – aschepler May 18 '22 at 15:49
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    Yes, thank you. – Mailbox May 18 '22 at 15:52
  • This is the same as saying that $n\to \infty$ implies $\frac{n}{x}\to\infty$. – John Joy May 23 '22 at 14:45

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