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[Community Question] Calculus: Why $\left( 1 + \frac{1}{x} \right)^x = 1 + x\frac{1}{x} + \frac{x(x-1)}{2}\cdot \frac{1}{x^2} + \dots $

One of our user asked:

How does one get the following expansion :

$$\lim_{x \to \infty} \left( 1 + \frac{1}{x} \right)^x = \lim_{x \to \infty} \left( 1 + x\frac{1}{x} + \frac{x(x-1)}{2}\cdot \frac{1}{x^2} + \dots \right)$$


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