When the function tends to infinity, there is a limit and a derivative function exists. Does the limit have to be 0 at infinity?

Of course, the derivative is not necessarily 0.

For example:

f(x)=[sin(x^2)]/x

f'(x)=[2x^2cosx^2-sinx^2]/x^2=2cos(x^2)-sin(x^2)/x^2

When x 2 = 2kπ, f' (x)->; 2。

Infinity in mathematics

1, geometry and topology

Infinite dimensional space is often used in geometry and topology, especially classification space, that is, Eilenberg? Common examples of MacLane space are infinite dimensional complex projective space K(Z, 2) and infinite dimensional real projective space K(Z/2Z, 1).

2. Irregular fragment shape

The fractal structure can be enlarged repeatedly, and the fractal can be infinitely enlarged, but it will not become smooth and still maintain its original structure. The perimeter of fractal is infinite, some areas are infinite, but some areas are finite. For example, the Koch curve is an example of infinite perimeter and limited area.

3. There is no infinite mathematics.

Leopold Kronecker doubted the concept of infinity, and also doubted that mathematicians used infinite methods in 1960s and 1960s. This skepticism forms a mathematical philosophy called finitism, which belongs to an extreme form of mathematical structuralism and mathematical intuitionism.