[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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Calculus: Intersection of hyperplanes in $\mathbb{R}^5$

Consider the $4$ hyperplanes in $\mathbb{R}^5$ given by the equations $$x_1-x_2+x_3+x_4+2x_5=0$$ $$2x_1-x_2+6x_3+2x_4+6x_5=0$$ $$3x_1-3x_2+3x_3+4x_4+7x_5=0$$ $$x_1+x_2+9x_3+x_4+6x_5=0$$ Let $V \leq \mathbb{R}^5$ be the intersection of these hyperplanes. Find a basis for $V$ .

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