Let the function f(x) be defined in the eccentric neighborhood of point x0. If there is a constant a, for any given positive number,

No matter how small, there is always a positive number.

So when x satisfies the inequality,

, the corresponding function values f(x) all satisfy the inequality.

Then the constant a is called the function f(x), when

Time limit, recorded as

Extended data

Four Algorithms of Function Limit

Assuming that f(x) and g(x) exist in the same independent variable process, their limits of sum, difference, product and quotient (the function as denominator and its limit value are not equal to 0) also exist, and the limit value is equal to the sum, difference, product and quotient of the limits. Multiplying a non-zero constant by a function does not change the existence of function limits.

Related theorem: pinch theorem

Let L(x), f(x) and R(x) satisfy L(x)≤f(x)≤R(x) in an eccentric neighborhood or infinite neighborhood during the process of independent variable change, and the limits of L(x) and R(x) exist and are equal during the process of independent variable change, then the limit of f(x) is also in the process of independent variable change.