Limit algorithm

Theorem 1The sum of two infinitesimals is infinitesimal

Theorem 2The product of a bounded function and an infinitesimal is an infinitesimal
Corollary
(1) The product of a constant and an infinitesimal is an infinitesimal
(2) The product of a finite infinitesimal is an infinitesimal

Theorem 3If limf(x)=A and limg(x)=B
then:
3.1 lim[f(x)± g(x)] = limf(x) ±limg(x)=A±B
3.2 lim[f(x) *g(x)] = limf(x) * limg(x) = A * B
3.3 If B is not equal to 0, then lim[f(x)/g(x)] = limf(x) / limg(x) = A / B
Corollary:
(1) If limf(x) exists and c is a constant, then
lim[cf(x)] = climf(x)
(2) If limf(x) exists and n is a positive integer, then
lim[f(x)]^n = [limf(x)] ^n

Theorem 4Set number sequence {xn} and {yn}, if
limxn = A (n->infinity)
limyn = B (n->infinity)
4.1 lim[xn± yn] = limxn ±limyn=A±B
4.2 lim[xn* yn] = limfxn * limyn = A * B
4.3 If B is not equal to 0, then lim[xn/yn] = limxn / limyn = A / B

Theorem 5 If f(x) >= g(x), limf(x) = A, limg(x) = B, then A>=B