The proof of two important limits of higher numbers

The proof of two important limits of higher numbers

SiNx / X → 1, (x → 0) is proved by the pinch criterion, using TaNx = SiNx / cosx > x > SiNx (in the first quadrant of the unit circle)
Note that when x → 0, cosx → 1; then we can get SiNx ~ x, X → 0 from the pinch criterion;
The other is proved by the theorem that a monotone bounded sequence of numbers must have a limit. Firstly, it is shown that the sequence of numbers is increasing, and then it must be less than 3 through expansion and contraction. Then a value E = 2.718281828459045 is given directly
The process of scaling is troublesome, involving index and binomial expansion