## Learning Calculus II: Limits’ laws for functions and related theorems

**2. Limits’ properties and related theorems **

We can see that it’s really hard to find the limit of a complex function by using formal definition directly. Hence, here is some limits’ properties and theorems that can be useful in finding limit. Firstly, let’s rewrite definition of limits:

**Definition 11**

*Let be a function defined on an open interval that contains , except possibly at . We say the limit of as approaches is , and we write*

if for every number there is a number such that:

if for all and then .

** 2.1. Limit laws for functions **

**Theorem 12 (Limit laws for functions)**

*Let be functions defined on interval that contains . Suppose have limit at . Then*

- (a) has limit at . We can write
- (b) has limit at . Similarly for . We can write
- (c) If is a real number, then has a limit at . We can write
- (d) has limit at . We can write

*Proof:* (a) It suffices to prove , i.e. according to definition 11 for any there is so for all . We need to use given condition that and , i.e. for any there exists so that for all and for all . Note that

Hence, we can pick so that . This will follow for all . Thus, (a) is proven according to definition 11.

## Learning Calculus I: Definitions of limits

**1. Definition of limits **

** 1.1. Definition of limits of function **

For me, the precise definition of limits is a bit hard to understand so I decided to reconstruct the whole thing by writing some of my guesses on why did mathematicians write the definition in this way. By the way, French mathematician, Augustin-Louis Cauchy (1789-1857) is the one who introduced into definition of limits.

Limit, the notation is understood vaguely as: as gets closer to , gets closer to a number . What does it mean by ‘get closer’? It is easier to understand if we imagine lies on a number line and suppose we pick an arbitrary point on the line. If we apply the term ‘ gets closer to ‘, we can understand it as distance from to on the number line decreases. Its distance is so saying gets closer to is the same as saying gets closer to . Note that for any distance , there are two possible on the number line that can satisfy that distance. What we care about is the distance from to not the position of , so using ‘ gets closer to ‘ is better than ‘ gets closer to ‘. Similarly to and we get a new definition:

Note that we can get as close to as we want by choosing appropriate , but we can’t guarantee the same thing for since depends on . Thus, firsly, in order for to be able to approach we need to be as small as we want. It is equivalent to saying:

For any there exists some so .

Secondly, we need the definition to say approaches as approaches . This means for small enough , must be bounded by some for all . For better understanding more about this argument, let’s assume the contrary, then as gets closer to , still does not have a bound for all . Say we find at has . We consider all in and follow that there must be an so and according to the assumption. Next consider all in and from our assumption, there exists so for . This keeps going and we can see that as gets closer to , gets larger, which contradicts to our aim that must approaches . Thus, we got the second part of the definition: (more…)

## LA I: Identify a list of vectors linearly dependent or linearly independent, a vector space infinite-dimensional or finite-dimensional

Inspired from some good exercises from Chapter 2A in the book *Linear Algebra Done Right* by Sheldon Axler.

First, let’s write out definition of linearly (in)dependent list:

**Definition 1 (Linearly independence)**

*A list of of vectors in is called*

**linearly independent**if the only choice of so is .The definition itself can be applied for linearly (in)dependent test. Also from this, we find

**Proposition 2**

*Every vector in the span of linearly independet list can be represented*

*uniquely*as .*Proof:* Indeed, assume the contrary that with not all equal to . Hence,

with not all equal to . This contradicts to our definition.

This proposition doesn’t help us much in testing linearly (in)dependence. But the below proposition, which is also deduced from the definition, has better application in testing, simply by looking at the list of vectors:

**Proposition 3**

*If some vector in a list of vectors in is a linear combination of the other vectors, then the list is linearly dependent, i.e. list that is not linearly independent.*

*Proof:* List has then , or there exists not all so . Hence, is linearly dependent.

## Reading a solution using algebraic integers of a HMMT number theory problem

I tried to attack the following problem but eventually gave up.

**Problem 1**

*(HMMT Ferb 2015 Team P9) Let and . Prove that*

*is an integer and find its remainder upon division by *

The only idea I have is to expand this expression, but it was too complex to do. So I decided to read the solution using algebraic integers. I’m new at algebraic number theory and I don’t know much about algebraic integers. My only reference is Problems from the book (PFTB), Chapter 9.

## Yellowstone Permutation – St. Petersburg MO 2015, grade 9, 2nd round

This problem was given to the St. Petersburg Olympiad 2015, 2nd round, grade 9 and is known as Yellowstone Permutation. You can see here for the article about this sequence and here for the OEIS entry.

**Problem. **(Yellowstone Permutation) A sequence of integers is defined as follows: and for , is the smallest integer not occurring earlier, which is relatively prime to but not relatively prime to Prove that every natural number occurs exactly once in this sequence.

*M. Ivanov*

Here is my proof for this interesting problem. Another proof can be found in the article given link above.

## The normal approximation to the binomial distribution

A coin is flipped times with probability of getting heads is . This is a binomial approximation . The Year 12 MATH B textbook gives an normal approximation to this: where is the mean (or expected value) and is the standard deviation. Since the textbook doesn’t give a proof for this so I will go and prove.