1, the idea of limit is the basic idea of calculus, and a series of important concepts in mathematical analysis such as continuity, derivative and definite integral of functions are defined by means of limit.

2. Mathematical analysis can solve many problems that elementary mathematics can't solve (such as finding instantaneous velocity, curve arc length, curved edge area, surface volume, etc.). It is precisely because it adopts the extreme thinking method.

Sometimes when we want to determine a quantity, we must first determine its approximation instead of the quantity itself, and the determined approximation is not just one but a series of more and more accurate approximations; Then, by examining the trend of this series of approximations, the exact value of this quantity is determined. This is the way of thinking by using the limit.

Extended data

The germination of limit thought can be traced back to ancient Greece and China during the Warring States Period, but the concept of limit first appeared in arithmetica infinitorum by Wallis, and Newton explicitly used the word limit in Mathematical Principles of Natural Philosophy.

But as late as the second half of18th century, D'Alembert and others realized that calculus would be perfect if it was based on the concept of limit. Cauchy first gave a descriptive definition of limit, and then Weierstrass gave a strict definition of limit (ε-δ and ε-N definitions).

From then on, all kinds of limit problems have practical criteria, making limit theory a tool and foundation of calculus.

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