The method of judging the existence of function limit can be realized by mathematical definition or sequence limit. These two methods will be introduced separately below.

1. Mathematical definition:

The definition of function limit is needed to judge whether a function has a limit at a certain point. For the function f(x), when x approaches a certain point c, if there is a constant L, it makes ε > for any given value; 0, with corresponding δ > 0, so when 0

This definition means that when x is close enough to C, the value of f(x) will be closer to L. If this condition is met, the limit of the function at point C exists.

2. Sequence restriction method:

In some cases, the existence of function limit can be judged by considering the sequence limit of function. The specific method is as follows:

Firstly, a point C can be approximated by the sequence {x_n}, thus ensuring that the sequence satisfies lim (n →∞) x _ n = C.

Then calculate the corresponding function value sequence {f(x_n)} and its limit lim(n→∞)f(x_n).

If this limit exists and has nothing to do with C, it can be judged that the limit of the function exists at point C, and the limit value is equal to lim(n→∞)f(x_n).

It should be noted that this method may not be suitable for infinite and infinitesimal limits.

When judging the existence of function limit, we can use one or a combination of these two methods to verify it. No matter which method is adopted, strict mathematical proof is needed to support the results, and various possible situations and boundary conditions are considered.

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. Such as the uniqueness of the function limit (if the limit exists, the limit at this point is unique).