Given any x, how can I calculate $\lim_{n \to \infty} (1 + \frac{x}{n})^n$ ?
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This is kind of a difficult question for somebody who is learning about limits. Of course it will be very beneficial for you to understand the proof. It is a difficult because the term under power converges to 1 ... yet it is always larger than 1 and we raised it to large power ... so, it kind of balances and the limit exists. – Salcio Apr 19 '23 at 12:27
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1This depends entirely on what you know about the number $e$ and how it is defined. It is not uncommon for this to be the defining expression for $e^x$ which you can then show is equivalent to the others. Other people may have been taught $\text{exp}(x) = 1+\frac{x}{1}+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots+\frac{x^k}{k!}+\dots$ and they can then show that is equivalent to your expression. Others still might have been taught that $e$ was the unique positive number such that the derivative of $e^x$ is equal to itself. Context is hugely important here, and the exact answer varies. – JMoravitz Apr 19 '23 at 12:28