The independent variable tends to infinity: note that x→∞ in the function limit refers to ∣x∣→ ∞

The independent variable tends to a finite value: here x tends to x0 and is not equal to x0. The limit value is only related to the derivative value in the decentered neighborhood of x=x0

The left and right limits exist and are equal

The boundedness of the limit of a sequence: xn must be bounded to converge, but bounded does not necessarily mean it will converge.

Local boundedness of function limit: if limx→x0f(x) exists, Then f(x) is bounded in the decentered neighborhood of point x0.

If A>0(<0), then f(x)>0(<0) in the centroid neighborhood

If f(x)≥0 (≤0) in the centroid neighborhood, then A≥0 (≤0); Or f(x)>0 in the centroid neighborhood, it can also be deduced that A≥0

Local sign preservation of continuous functions: If the function f(x) is defined in a certain decentered neighborhood of x=a point 0<∣x−a∣<r, f(x) is continuous at point x=a, and f(a)>0 (or <0), then there exists a certain (solid) neighborhood∣x−a∣<δ, For all x in the decentered neighborhood, f(x)>0 (or <0) is always present.

A sequence that is monotonically increasing and has an upper bound must have a limit. A sequence that is monotonically decreasing and has a lower bound must have a limit.

The sum of a finite number of infinitesimals is still infinitesimal The product of a finite number of infinitesimals is still infinitesimal The product of an infinitesimal quantity and a bounded quantity is still infinitesimal

The product of two (can also be extended to finite) infinite quantities is still an infinite quantity

The sum of an infinite quantity and a bounded variable is still an infinite quantity

Infinity must be unbounded, but unbounded does not necessarily mean infinite.

In the same limit, if f(x) is infinite, then 1/f(x) is infinitesimal; Conversely, if f(x) is infinitesimal and f(x) is not equal to 0, then 1/f(x) is infinite.

Corollary 1: The limiting non-zero factor can be found first Corollary 2: If lim f(x)/g(x) exists and lim g(x)=0, then there must be lim f(x)=0

Corollary 3: If lim f(x)/g(x) =A (A is not 0, if limf(x)=0, then there must be lim g(x)=0

Continuous (continuous ± discontinuous = discontinuous, the rest are not necessarily) Differentiable (differentiable ± non-differentiable = non-differentiable, the rest are not necessarily different) Series (convergence ± divergence = divergence, the rest are not necessarily)

You can change the multiplication and division factors at will

Additive substitution: the ratio of the two addition terms is not negative one Subtraction substitution: two subtraction terms are not equivalent

If f(x) is differentiable to order n, the use of Lópida's rule can only occur up to n−1 order of f(x). If f(x) has an n-th order continuous derivative, using L'Hobida's rule, it can appear to the n-th order of f(x).

Lópida

Taylor formula

Equivalent infinitesimal substitution

Lópida

The numerator and denominator are divided by the highest order term (find the boss)

Become 0 to 0, or infinity to infinity

Pass differentiation into 0 to 0 type (applicable to fractional difference)

Rationalization of radical expressions (applicable to radical differences)

When there is no denominator in the function

rewritten in exponential form

Make up the second important limit

This is the power function form, rewritten as ln in the exponential form e