ln( 1+e^x)-x (x→+∞)

=ln( 1+e^x)-ln(e^x) (x→+∞)

=ln[( 1+e^x)/(e^x)] (x→+∞)

=ln 1

When x → +∞, Ln [1+e (-x)] is also →ln 1.

Therefore, ln 1 limit is one of the most basic concepts in other branches of calculus and mathematical analysis, and the concepts of continuity and derivative are defined by it. It can be used to describe the changing trend of element properties in a sequence when the exponent of a sequence is getting larger and larger, and it can also be used to describe the changing trend of the corresponding function value when the independent variable of a function approaches a certain value.

The thinking function of extreme thinking

Extreme thoughts are widely used in modern mathematics and even physics, which is determined by its inherent thinking function. The thought of limit reveals the unity of opposites between variables and constants, infinity and finiteness, which is the application of the law of unity of opposites of materialist dialectics in the field of mathematics.

With the help of limit thought, people can know infinity from finiteness, change from invariability, curve from linear formation, qualitative change from quantitative change and accuracy from approximation.