Comparison method is the most basic method to prove inequality, including difference comparison method and quotient comparison method. The basic idea is to change the formula that is difficult to compare into the difference between it and 0 or the comparison size between it and the quotient of 1. When the two ends of the proved inequality are parts (or fractions), it is often used for difference comparison, and when the two ends of the proved inequality are products (or power exponents), it is often used for quotient comparison. Solution: Let the function f (x) = e x and g (x) = x+ 1.

For the function f (x) = e x, it is a natural exponential function, the definition domain is all real numbers, the function is monotone increasing function in the definition domain, and the range is: [0, +∞). The schematic diagram of the image is as follows:

2. For the function g(x)=x+ 1, it is a linear function, and its domain and value domain are real numbers. In the domain, the function is increasing function, and the diagram of the image is as follows.

3. As can be seen from the figure, the function g(x)=x+ 1 is below the function F (x) = e x, and the two have an intersection point (0, 1), so there is:

f(x)>=g(x)

That is, e x > =x+ 1, which holds.

The first is the definition of limit, which is rarely used, but it can also be used to find the limit. Two important laws are defined: pinch and monotonicity. Pinch theorem correctly choosing "limit" is an extremely important basic concept in higher mathematics. Concepts such as derivative, definite integral, generalized integral and curve asymptote are all based on limit, and limit is an important tool for learning calculus.