$$
G(x)=int_{a}^{x}f(t)dt Rightarrow G’(x)=f(x) Rightarrow frac{d}{dx}int_{a}^{x}f(t)dt=f(x) Rightarrow int f(x)dx=int_{a}^{x}f(t)dt+C
$$

$$
lim_{x to a}f(x)=g(x)=0 Rightarrow lim_{x to a} frac{f(x)}{g(x)}=lim_{x to a}frac{f’(x)}{g’(x)}
$$

$$
begin{aligned}
lim_{x to a} frac{f(x)}{g(x)}
=lim_{x to a} frac{frac{f(x)-0}{x-a}}{frac{g(x)-0}{x-a}}
=lim_{Delta x to 0} frac{frac{f(a+Delta x)-f(a)}{Delta x}}{frac{g(a+Delta x)-g(a)}{Delta x}}=frac{f’(a)}{g’(a)} (g’(a) not= 0)
end{aligned}
$$

$$
lim_{x to a} f(x)=0,g(x)=+infty Rightarrow lim_{x to a}f(x)=0,frac{1}{g(x)} = 0 Rightarrow lim_{x to a} f(x)g(x)=lim_{x to a}frac{f(x)}{frac{1}{g(x)}}
$$

$$
lim_{x to 0}f(x)=g(x)=0 Rightarrow lim_{x to 0}f^{g(x)}(x)=lim_{x to 0}e^{g(x)ln f(x)}
$$

$$
sum_{i=0}^{infty}a_i = C Rightarrow lim_{i to infty} left|frac{a_{i+1}}{a_i}right| < 1
$$