Urgent! If the function f (x) is differentiable at x = 0, and f (0) = 0 x tends to 0: then find the limit f (x) / x =?

Urgent! If the function f (x) is differentiable at x = 0, and f (0) = 0 x tends to 0: then find the limit f (x) / x =?

According to the definition of limit:
Lim [f (x) - f (0)] / (x-0) = Lim [f (x) / x] = f '(0) when x → 0

Piecewise function f (x) ln(1+ax^3)/(x-arcsinx) ,x0 When asked what the value of a is, f (x) is continuous at x = 0

a=-1; If f (x) is to be continuous at x = 0, it only needs to have: [x - > 0] limf (x) = f (0). For this problem, if f (x) is continuous at point 0, then it can be obtained that: [x - > 0 +] LIM (e ^ (AX) + x ^ 2-ax-1) / X * sin (x / 4) = 6; [x->0-]lim ln(1+ax^3)/(x-arcsinx)=6; For the first equation, sin (x / 4) can be X

There is a problem in understanding the uniqueness of function limit When the limit of F (x) exists at a certain point, why must it be unique? Can't there be another point with a limit?

The uniqueness of limit means that there can only be one limit at a certain point. Of course, there can be a limit at another point, and its limit is also unique. Many points may have limits, but once this point is determined, the limit is also determined, and there can be no two limits at the same point
I hope it can help you. If you solve the problem, please click the "select as satisfactory answer" button below,

What is the definition of neighborhood? A more popular definition

Point centered and ε The whole of the inner points of a circle with a radius, that is, the set, is called the neighborhood of the point, and the point is called the center of the neighborhood, which is the radius of the neighborhood

What is neighborhood? It is a concept in higher mathematics

Neighborhood
Any open interval centered on a is called the neighborhood of point a and is recorded as u (a)
set up δ Is any positive number, then in the open interval (a- δ, a+ δ） Is a neighborhood of point a, which is called the neighborhood of point a δ Neighborhood, denoted as u (a), δ), I.e. U (a), δ)= {x|a- δ

The definition of neighborhood is that any open interval centered on a is called the neighborhood of point A. how should this "center" be understood? Must it be a symmetric center

The field is (A-C, a + C) and the radius is C, so a is the midpoint