lim[x→x0]f(x)=A→0

The algebraic sum of a finite number of infinitesimal quantities is still an infinitesimal quantity

The product of a finite number of infinitesimal quantities is still an infinitesimal quantity

The product of a constant and an infinitesimal quantity is an infinitesimal quantity

The product of a bounded variable and an infinitesimal quantity is an infinitesimal quantity

Higher order infinitesimal/lower order infinitesimal

infinitesimal of the same order

Equivalent to infinitesimal

sinx~x

arcsinx~x

tanx~x

arctanx~x

e^x-1~x

ln(1 x)~x

1-cosx~1/2x²

(1 αx)^β~αβx

(Example: when x→0, 1/x, 1/x², 1/sinx, 1/tanx are all infinite quantities)

(Example: when x→∞, x², e^x, ln(x 1) are all infinite quantities.)

In the same process of change, the reciprocal of infinity is infinitesimal

The reciprocal of infinitesimals that is always not equal to zero is infinity

Eliminate terms with a denominator of 0 to facilitate calculations

f(x) is defined at point x0

lim[x→0]f(x) exists

lim[x→x0]f(x)=f(x0)

f(x) is not defined at point x0

lim[x→x0]f(x) does not exist

lim[x→x0]f(x)≠f(x0)

Discontinuity points can be removed: if lim[x→x0-]f(x) and lim[x→x0 ]f(x) both exist, and lim[x→x0-]f(x)=lim[x→x0 ] f(x), but f(x) is not defined at x0. Then the point x0 is called the discontinuous point of f(x).

Jump discontinuity point: If both lim[x→x0-]f(x) and lim[x→x0 ]f(x) exist, but lim[x→x0-]f(x)≠lim[x→x0 ]f (x), then point x0 is called the jump discontinuity point of function f(x).

Infinite discontinuity point: If limf(x)=∞, or lim[x→x0-]f(x)=∞, or lim[x→x0]f(x)=∞, then the point x0 is called the function f(x ) of infinite discontinuities.

Oscillation discontinuity point: If when x→x0, the function value f(x) changes between two different numbers infinitely, then the point x0 is called the oscillation discontinuity point of the function f(x).