1, when x tends to 0, sinx is equivalent to X. This substitution is very common in mathematical operations such as limit, derivative and integral. For example, when x tends to 0, sin (x 2) is equivalent to x 2.

2. when x tends to 0, tanx is equivalent to X. this substitution is usually used to deal with mathematical expressions with tangent functions. For example, when x tends to 0, tan (x 3) is equivalent to x 3.

3. arcsin(x) is equivalent to x when x tends to 0. This substitution is usually used to deal with mathematical expressions containing arcsine functions. For example, when x tends to 0, arcsin(x^2) is equivalent to x 2.

4. when x tends to 0, arctan(x) is equivalent to X. This substitution is usually used to deal with mathematical expressions containing arctangent functions. For example, arctan(x^3) is equivalent to x 3 when x tends to 0.

5. infinitesimal substitution of bounded function when x tends to infinity. For example, when x tends to infinity, arctan( 1/x) is equivalent to π/2-π/(2x).

Key points of equivalent infinitesimal substitution;

1. Usage scenario: Equivalent infinitesimal substitution is mainly used to find various limits, especially the limits of complex variable functions. For example, when solving the limit in the form of "0/0" or "∞/∞", it is often necessary to solve it by equivalent infinitesimal substitution.

2. Principle: The main principle of using equivalent infinitesimal substitution is to substitute those items that are infinitely close to zero near a specific point. The purpose of this is to simplify the calculation, because simple functions are easier to handle than complex functions.

3. Precautions: Although equivalent infinitesimal substitution is a powerful tool, there are also some precautions. First of all, not all terms can be replaced by equivalent infinitesimal. Secondly, when replacing, we need to be careful to ensure the accuracy of the results. Finally, the concept of infinitesimal and its properties are the basis of understanding and applying equivalent infinitesimal substitution, which must be deeply understood.

4. Summary: Equivalent infinitesimal substitution is an important concept in calculus, which is mainly used to deal with limit problems. We can simplify the calculation by replacing complex functions with simple functions in the neighborhood of specific points. However, when using equivalent infinitesimal substitution, we should pay attention to its applicable conditions and possible errors.