Conditions and details of polar coordinate substitution for solving the limit of multivariate functions with high numbers

Preface

When finding the multiple limit of a function of two variables , Polar coordinate substitution is called " Universal solution ", Many textbooks directly use polar coordinate substitution without proof . But some people think this method " loosely ", Can't meet " Any path approaches ".

Why does someone say that polar coordinate substitution is omnipotent ? Why does someone say this thing is not rigorous ?

There is nothing rigorous or not , Yes, that's right. , Wrong is wrong , Today's article will explain everything to you .

Read the article carefully , You can fully understand and master the general method of solving the double limit , From then on, it is as natural to seek the weight limit as to drink water .

Where is the problem ?

Many teachers said , Use polar coordinate substitution , Transform the limit of binary function into the limit of unary function , Very convenient to solve .

Let's first look at such an example :

example

Explain

that , What's the problem ?

Find the answer in the definition

Many people think that polar coordinate substitution is actually approaching along a straight line , Not approaching along the curve , dissatisfaction " Approach along any path ", So it's going to go wrong .[1][2]

  Liu Jiying . Using polar coordinate substitution method to find the conditions that the double limit should meet [J]. JOURNAL OF LIAONING TEACHERS COLLEGE ( Natural science edition ),2006(01):3.

  Zhihu article 《 The strongest solution of the double limit —— Polar coordinate substitution ！！！》 Comment area

Is that really the case ? What do you mean " Follow any path "?

This statement itself is very vague . We can't list all the curve paths , You can't find the weight limit directly, can you .

Let's take a closer look at the definition of heavy limit .

  Definition [3]

Conditions for use of polar coordinate substitution

The discussion just now has obtained the conditions for the use of polar coordinate substitution ：

Obviously, as long as this condition is met , that x You can take any point in the open circle .

The detailed proof is shown in the figure below :[4]

  Xuzhaochun . Thinking about using the polar coordinate transformation method to solve the double limit [J]. Technology Information ,2009(31):243.

Summary

Now let's take a look at the example at the beginning , What is the problem ?

Misinterpretation

Exercises

practice 1

Methods to summarize

First step , Polar substitution , Simplification .

The second step , Check Can you take any value , If you can , Get the answer ; If not , Take the special curve and get that the heavy limit does not exist .

Reference resources

1. ^ Liu Jiying . Using polar coordinate substitution method to find the conditions that the double limit should meet [J]. JOURNAL OF LIAONING TEACHERS COLLEGE ( Natural science edition ),2006(01):3.
2. ^https://zhuanlan.zhihu.com/p/143541324
3. ^ Xue Yumei , Yuan Jia et al . Mathematical analysis of Engineering ( last of two or three volumes ). [M]. Beijing University of Aeronautics and Astronautics Press , 2020.
4. ^ Xuzhaochun . Thinking about using the polar coordinate transformation method to solve the double limit [J]. Technology Information ,2009(31):243.

  From the original ——

When finding the double limit , How to use polar coordinate substitution ? - You know