like $\lim_{x \to 0} \frac{\sin x}{x}$ Rendering
$$
\begin{eqnarray}
\lim_{x \to 0} \frac{\sin x + x^2 + x^3}{x} & = & \lim_{x \to 0} \frac{\sin x }{x} + \lim_{x \to 0} \frac{x^2 }{x} + \lim_{x \to 0}\frac{x^3}{x}
\\ & = & \lim_{x \to 0}\frac{\sin x}{x}
\\ & = & 1
\end{eqnarray}
$$
Fibonacci number column$A_n=A_{n-1}+A_{n-2}$,The ratio of the two items has gradually converged to the golden segmentation ratio
$$\lim_{n\to \infty}\frac{A_{n-1}}{A_n}=\frac{\sqrt{5}-1}{2}.$$
Factor decomposition
$$\begin{split}(x−1)(x−3)&=x^2−4x+3 \
&=x^2−4x+4−1 \
&=(x−2)^2−1
\end{split}
$$
Dili Krey function
$$
D(x)=
\begin{cases}
1,& x \in Q \
0,& x \notin Q
\end{cases}
$$
Gaussian formula
$$
\iiint_{\Omega}\left(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\right) d v=\iint_{\Sigma} P d y d z+Q d z d x+R d x d y
$$
Physical formula
- Newton’s first law: $\sum \vec{F}_{i}=\frac{\mathrm{d} \vec{v}}{\mathrm{d} t}=0$
- Newton’s second law: $\vec{F}=\frac{\mathrm{d} m}{\mathrm{d} t} \vec{v}+m \frac{\mathrm{d} \vec{v}}{\mathrm{d} t}=\frac{\mathrm{d} m}{\mathrm{d} t} \vec{v}+m \vec{a}=\frac{\mathrm{d} m}{\mathrm{d} t} \vec{v}+m \frac{\mathrm{d}^{2} \vec{r}}{\mathrm{d} t^{2}}$
- Newton’s third law: $\overrightarrow{F_{12}}=-\overrightarrow{F_{21}}$
- Conservation: $E=mc^2$
Chemical formula
Ion reaction and precipitation: $\ce{SO4^2- + Ba^2+ -> BaSO4 v}$
Biological formula
Photosynthesis
Van Demon ranked
$$D_{n-1}=\left|\begin{array}{cccc}
1 & 1 & \dots & 1 \
x_{2} & x_{3} & \dots & x_{n} \
\vdots & \vdots & & \vdots \
x_{2}^{n-2} & x_{3}^{n-2} & \dots & x_{n}^{n-2}
\end{array}\right|=\prod_{2 \leq j<i \leq n}\left(x_{i}-x_{j}\right)$$