How to Solve 5000 Problems in Mathematical Analysis with Demidovich's Book Mathematical analysis is a branch of mathematics that deals with the study of functions, limits, derivatives, integrals, series, and other topics. It is a challenging subject that requires a lot of practice and problem-solving skills. One of the most
famous and comprehensive books on mathematical analysis is Problems in Mathematical Analysis by Boris Demidovich, a Russian mathematician and educator. This book contains over 5000 problems of various levels of difficulty, covering all the main topics of
mathematical analysis.

However, finding the solutions to these problems can be difficult, especially for students who are learning mathematical analysis for the first time. There is no official solution manual for Demidovich's book in English, and the existing ones in other
languages are based on different editions and numbering of the problems. Therefore, it would be useful to have a guide that explains how to solve these problems step by step, using clear and concise methods.

Solucionario Demidovich 5000 Problemasl
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In this article, we will provide such a guide, based on the latest edition of Demidovich's book published by Mir Publishers in 1986. We will show you how to solve some of the most representative and interesting problems from each chapter of the book,
using examples, diagrams, formulas, and explanations. We will also give you some tips and tricks on how to approach these problems and avoid common mistakes. By following this guide, you will be able to master mathematical analysis and solve any problem
that Demidovich throws at you.

Let's get started!


Chapter I: Introduction to Analysis
In this chapter, we will review some basic concepts and definitions of analysis, such as functions, graphs, limits, continuity, and infinite quantities. We will also learn how to find the limits of various expressions and functions using algebraic and
graphical methods. We will also study some important properties and theorems of limits and continuity, such as the squeeze theorem, the intermediate value theorem, and the extreme value theorem.

Some of the problems in this chapter are:


Find the domain and range of a given function.
Sketch the graph of a given function and identify its asymptotes, zeros, extrema, and points of discontinuity.
Find the limit of a given expression or function as x approaches a finite or infinite value.
Determine whether a given function is continuous or discontinuous at a given point or on a given interval.
Use the definition of continuity to prove or disprove that a given function is continuous.
Use the properties and theorems of limits and continuity to solve various problems involving inequalities, equations, and functions.

For example, let's look at problem number 1 from section 3:

Problem 1. Find

\[\lim_x\to 0\frac\sin xx.\]



To solve this problem, we can use the following trigonometric identity:

\[\sin x=x-\fracx^33!+\fracx^55!-\cdots.\]

Dividing both sides by x, we get:

\[\frac\sin xx=1-\fracx^23!+\fracx^45!-\cdots.\]

As x approaches zero, all the terms except the first one become negligible. Therefore, we can write:

\[\lim_x\to 0\frac\sin xx=\lim_x\to 0\left(1-\fracx^23!+\fracx^45!-\cdots\right)=1.\]

Hence, the answer is 1.
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