Let f:(a, +∞)→R be a real function of one variable, and a ∈ R. If is for any given ε >; 0, there is a positive number x, so that for a suitable inequality x >；; All x of x, the corresponding function value f(x) satisfies inequality. │ f (x)-a │
The definitions of function limits can be divided into x→∞, x→+∞, x→-∞, x→Xo, ε-δ, which are common in the proof of known limit values. Mastering such proof is of great benefit to beginners. Taking the limit of X → Xo as an example, the definition of f(x) at the point Xo with limit is: for any given positive number ε (in any case, | x-x. | < When δ, the corresponding function values f (x) all satisfy the inequality: | f (x)-a | < ε, then the constant A is called the limit of the function f(x) when x → X, and the key to the problem is to find the one that meets the definition requirements. In this process, some inequality techniques, such as scaling, will be used. 1999 postgraduate exam questions, which also directly examines the examinee's mastery of the definition. See appendix 1 for details. Rational application of limit properties of functions. The common properties of function limit are uniqueness, local boundedness, order preservation, algorithm of function limit, composite function limit and so on. For example, the uniqueness of the function limit (if it exists, the limit of this point is unique).
The limits of some functions are difficult or difficult to be obtained directly by limit algorithm, and need to be determined first. Here are several commonly used theorems to determine the limit of a sequence. 1. pinch theorem: (1) when x∈U(Xo, r) (this is the centripetal neighborhood of Xo, so you can't type a symbol), there is g (x) ≤ Xo=A, h (x)->; Xo=A, then the limit of f(x) exists and is equal to A, which can not only prove the existence of the limit, but also find the limit, mainly by scaling method. 2. Monotone Bounded Criterion: Monotone increasing (decreasing) sequences with upper (lower) bounds will converge. When using the above two items to find the function limit, we should pay special attention to the following points. First of all, we must prove convergence by monotone bounded theorem. 2. The key to applying the pinch theorem is to find the function with the same limit value, so that ε > the limit value tends to the same direction, thus proving or finding the limit value of the function. 0, N(ε) exists, so when n >; N, m> When n, there is | am-an |