When a function converges at a certain point, it means that when the independent variable tends to this point, the limit of its value is equal to the value of the function at that point. Isn't that continuous The relationship between function convergence and function continuity

The convergence of a function at a certain point means that when the independent variable tends to this point, the limit of the function value exists, which has nothing to do with the value of the function at this point. This point can even be undefined, and whether there is a limit can be investigated. If a function is continuous at a certain point, it must converge, and the left limit and right limit at this point are equal to the value of the function

In higher mathematics, the limit value of a continuous function is equal to its function value. Why is the derivative value of a continuous function not its function value?

The limit of continuous function is the limit of function expression
The derivative of continuous function is the limit of [f (x2) - f (x1)] / (x2-x1),
The geometric meaning of the derivative is the slope of the tangent at this point

A necessary and sufficient condition for the existence of function limit when independent variable tends to finite value

Left and right limits exist and are equal

Definition of function limit when independent variable tends to infinity
The definition of function f (x) converging to a when x → - ∞

LIM (x → - ∞) f (x) = a for any given ε > 0, there is always x > 0 such that for any x

What is the difference between the limit of function when the independent variable tends to infinity and when the independent variable tends to a certain value?

When the independent variable tends to infinity, the limit of the function may not exist. When it tends to a certain value, it may be the function value of the function at the change point, it may be the limit value, or it may not exist

Does a point have a neighborhood?
The | x | 0 of heartless neighborhood

Of course, only a point has a neighborhood. And this point is not only limited to the coordinate axis, but also can be a point in two-dimensional and three-dimensional space
Any open interval centered on point a is called the neighborhood of point A. if the point a is removed from the open interval, it is called the centreless neighborhood
Heartless neighborhood 0

How to find the left and right limits of X tending to type x0
What is the general idea of this kind of problem?
For example, the left and right limits of the following question,
F (x) = xsin (1 / 2) x x = 0 point

A:
Is f (x) = xsin (x / 2)?
When x tends to 0, X / 2 tends to 0 and sin (x / 2) tends to 0
So: xsin (x / 2) tends to zero
So: LIM (x → 0) xsin (x / 2) = 0

F (x) has definition, limit and continuity at point x0. What's the difference between these three concepts

A definition is a necessary condition for continuity. It has nothing to do with a limit
A limit means that the left and right limits are equal. It has nothing to do with having a definition, but it is a necessary condition for continuity
[function value with definition + limit + definition = this limit] = Continuous

ln (1/e^2)

Ln (1 / e ^ 2) = log (E) 1 - 1og (E) e ^ 2 = 0-2 * log (E) e = 0-2 * 1 = - 2 Analysis: ln is the logarithm based on e, because the division of true number equals the subtraction of logarithm, that is, the square of true number 1 divided by e equals the logarithm of log based on e minus the logarithm of log based on e, and because the power of true number can be used as the system before the logarithm is obtained

When a function converges at a certain point, it means that when the independent variable tends to this point, the limit of its value is equal to the value of the function at that point. Isn't that continuous The relationship between function convergence and function continuity