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The limit of binary functions.
In mathematics, the concept of limits is one of the important tools for studying functions. For binary functions, the definition and properties of their limits are similar to those of unary functions, but there are some peculiarities. In this article, we will introduce the definition, properties, and calculation methods of the limits of binary functions.
1. Definition of the limit of a binary function.
Similar to a univariate function, the limit of a binary function is a fixed number of values that the value of a function approaches when the independent variable approaches a certain value. The specific definitions are as follows:
Let the binary function f(x, y) be defined in some neighborhood of the point (x0, y0), if for any given positive number , there is a positive number δ such that when |x - x0|And |y - y0|"Time, there is |f(x, y) -a|The limit of the function f(x, y) at the point (x0, y0) is said to be a.
This definition can be understood as follows: in the neighborhood of a point (x0, y0), as long as x is close enough to x0, f(x, y) will be infinitely close to a, regardless of the value of y. Similarly, no matter what value x takes, as long as y is close enough to y0, f(x, y) will be infinitely close to a.
2. The limiting properties of binary functions.
1.Uniqueness: If the limit of the binary function exists, the limit value is unique.
2.Local boundedness: If the limit of the binary function exists, it is bounded within a certain neighborhood of the limit point.
3.Local order-preserving: If the limit of the binary function exists, the value of the function retains its original magnitude relationship within a certain neighborhood of the limit point.
3. Limit calculation method of binary function.
1.Direct substitution: For some simple binary functions, you can directly substitute the independent variables into the function expression to calculate the limit. For example: lim (x, y) 0, 0) (x 2 + y 2).
2.Entrapment method: By comparing the value of the function with the known entrapment function, the limits of some binary functions can be calculated. For example: lim (x, y) 0, 0) (x 2 + y 2) = 0.
3.Parametric method: For some parameter-related binary functions, the limit can be calculated by eliminating the parameter. For example: lim (t 0) [1 + t) (1 t)] e = 1.
4.Equivalent infinitesimal substitution: When calculating the limits of a binary function, complex function expressions can sometimes be simplified into expressions that are easy to compute by equivalent infinitesimal substitution. For example: lim (x, y) 0, 0) (sin x) x) (xy) = lim (x, y) 0, 0) (sin y) y) (xy) = e (-1).
5.Special Point Method: The limit is calculated by selecting a special point to substitute it into the function expression. For example: lim (x, y) xy (x 2 + y 2)) = 1 2.
6.Taylor Expansion: For some binary functions that are difficult to calculate, the limit can be calculated by expanding it into an easy-to-compute power series form. For example, lim (x, y) 0, 0) (sin x) x) (xy) can be calculated using the Taylor expansion to obtain the result e (-1).
In summary, the limit of binary functions is one of the important tools for studying functions. By grasping the definition, properties, and calculation methods of limits, we can better understand the properties and behavior of functions. In the subsequent learning, we will continue to learn more about the in-depth knowledge and application of binary function limits.