Definition: Sequence {a_n} When n approaches infinity, if there is a real number L such that the absolute value of the difference between a_n and L can be arbitrarily small, then L is called the limit of the sequence {a_n}.

Properties: uniqueness, boundedness, number preservation, pinch theorem

Definition: Function f(x) When x tends to a certain point a, if there is a real number L such that the absolute value of the difference between f(x) and L can be arbitrarily small, then L is called a function f(x) when x tends to The limit at time a.

Left limit and right limit

Infinitely Small and Infinitely Large

Condition: The function is continuous at a certain point or the limit form is simple

Application: polynomials, rational functions, etc.

Condition: Infinitive limit where the numerator and denominator are both 0

Steps: factoring, reduction, and substitution calculations

Condition: 0/0 type or ∞/∞ type infinitive limit

Steps: Find the derivative, substitute it into the calculation, and repeat the application until it can be directly calculated.

Condition: Limit calculation of complex functions

Steps: Expand the function into a Taylor series near a certain point, intercept appropriate terms, and calculate the limit

Condition: Two functions force a function, and the limits of the two functions are the same

Steps: Find the clamping function, prove the clamping relationship, and calculate the limit

Theorem: If the limit of a sequence or function exists, then the limit is unique

Theorem: If a limit of a sequence or function exists at a certain point, then the sequence or function is bounded near that point.

Theorem: If the limit of a sequence or function is greater than 0 (or less than 0), then the sequence or function maintains the same sign near the limit point

Theorem: Limit operations can be exchanged with addition, subtraction, multiplication and division operations.

Theorem: If the outer function is continuous at a certain point and the limit of the inner function exists at a certain point, then the limit of the composite function exists at that point.

Definition: When x approaches a certain point, if the ratio of one infinitesimal to another infinitesimal approaches 0, the former is said to be a higher-order infinitesimal of the latter.

Definition: When x approaches a certain point, if the ratio of one infinitesimal to another infinitesimal approaches infinity, the former is said to be the lower-order infinitesimal of the latter.

Definition: When x approaches a certain point, if the ratio of one infinitesimal to another infinitesimal approaches a non-zero constant, then they are said to be infinitesimals of the same order.

Definition: When x approaches a certain point, if the ratio of two infinitesimals approaches 1, then they are said to be equivalent infinitesimals.

Theorem: A sequence that is monotonically increasing (or decreasing) and has an upper (or lower) bound must have a limit.

Theorem: The necessary and sufficient condition for the convergence of the sequence {a_n} is that for any positive number ε, there exists a positive integer N such that when m,n>N, a_m a_n < ε

Theorem: The necessary and sufficient condition for the existence of the limit of a function at a certain point is that both the left limit and the right limit exist and are equal.

The definition of e: lim (1 1/n)^n The limit when n tends to infinity

Important limit: lim (sinx)/x The limit when x tends to 0

Important limit: lim (ln(1 x))/x The limit when x tends to 0

Important limit: lim (arctanx)/x The limit when x tends to 0

Theorem: If the limit of a function at a point exists and is equal to the function value at that point, then the function is continuous at that point

Definition: The derivative is the instantaneous rate of change of a function at a certain point and can be defined by limits

Definition: The definite integral is the limit of the area of ​​the curved trapezoid of a function on a certain interval, which can be defined by the limit.

Theorem: The convergence of a series can be determined by examining the limits of parts and sequences