Calculus/Limits

Introducing LimitsEdit

Let's consider the following expression:  

As n gets larger and larger, the fraction gets closer and closer to 1.

 

As n approaches infinity, the expression will evaluate to fractions where the difference between them and 1 becomes negligible. The expression itself approaches 1. As mathematicians would say, the limit of the expression as   goes to infinity is 1, or in symbols:  .

An interesting sequence is   As   gets bigger (in symbols  , we have smaller values of  , for

 

and so on. Clearly,   can't be smaller than zero (for if   we have that   is less than zero). Then we may say that  . Continuing with this sequence, we might want to study what happens when   gets near to zero, and later what happens with negative values of   going near to zero. Usually, the letter   is reserved for integer values, so we are going to redefine our sequence as  . If we take a sequence of values of  , say

 

We see that the respective values of   grows indefinitely, for

 
 
 
 
 

In this case, we might say that the limit of  , in words, the limit of   as   goes to zero from right (as the sequence of values of   goes to zero from the left in a graphic) diverge (or tends to infinity, or is unbounded, but we never say that it is infinity or equals infinity). Other case happens if we study sequences of values of   such that every element of the sequence is lower than zero, the sequence is increasing but never exceeding zero. One example of such sequence is:

 , with  
 , with  
 , with  
 , with  
 , with  

The values of   decrease without bounds. The we say that  , or that   tends to minus infinity when   goes to zero from left.

Basic LimitsEdit

For some limits (if the function is continuous at and near the limit), the variable can be replaced with its value directly:
 
For example,
 
and
  (with   not equal to 0)

Others are somewhat more complicated:
 

Note that in this limit, one may not immediately set   equal to   because this would result in the expression evaluating to
 

which is an undefined expression. However, one may reduce the expression by separating the terms into separate fractions (in this case,   and  ), which can be evaluated directly.

Right and Left Hand LimitsEdit

Sometimes, we want to calculate the limit of a function as a variable approaches a certain value from only one side; that is, from the left or right side. This is denoted, respectively, by   or  . If the left-hand and right-hand limits do not both exist, or are not equal to each other, then the limit does not exist.
The following limit does not exist:  
It doesn't because the left and right handed limits are unequal.
 
 

Note that if the function is undefined at the point where we are trying to find the limit, it doesn't mean that the limit of the function at that point does not exist; for an example, let's evaluate   at  .

Left-hand limit Right-hand limit
   

 

Therefore:  

Some Formal Definitions and PropertiesEdit

Until now, limits have been discussed informally but it shouldn't be all intuition, for we need to be sure of certain assertions. For an example, the limit

 

We have seen that the function decreases as   increases, but how do we guarantee that there isn't a value  , say

 

such that   is never smaller than  ? If there is such  , we might want to say that the limit is  , not zero, and we can't test every single possible value of   (for there are infinite possibilities). We must then find a mathematical way of proving that there isn't such  , but for that we need to define formally what a limit is.


Right Limits, Left Limits, Limits and ContinuityEdit

Let   be a real valued function. We say that

 

if for every   there is a   such that, for every   between   and  

 

where   is the absolute value of  .

This is the formal definition of convergence from the left. It means that for each possible error bigger than zero, we are able to find a interval such that for all   in that interval, the distance between the value of the function and the constant   is less than the error.

TODO: Graphics illustrating this.

In an analogous fashion, we say that

 

if for every   there is a   such that, for every   between   and  ,  .

And to finish the necessary definitions,

 

if

 

and

 .

Example:

 

This is a assertion that must be proved. First, lets study the behavior of   near zero;

 

where the arrow pointing right means implies. So, define the function

 

If  , then  . We have show how to find the delta in the definition of limit, showing that the limit of   as x tends to zero is zero.

In fact, for any real number  ,

 

Lets see how to construct a suitable function  .

 , then  

So,

 

implying that   makes   for any  .

Functions with the property that

 

are called continuous, and arise very naturally in the physical sciences; Beware that, against the intuition of most people, not every function is continuous.

Property of LimitsEdit

Property one: If  , then

 

for any constant  .
Proof: Construct the function   for  . Then

 

So

 

Then the limit of   is  , for the delta function of   is

 

QED.

TODO: Demonstrate main properties of limit (unicity, etc)

L'Hôpital's RuleEdit

L'Hôpital's Rule is used when a limit approaches an indeterminate form. The two main indeterminate forms are   and  . Other indeterminate forms can be algebrically manipulated, such as  .

L'Hôpital's Rule states if a limit of   approaches an intederminate form as   approaches  , then:
 

Example:  
Both the numerator and the denominator approach zero as   approaches zero, therefore the limit is in indeterminate form, so l'Hôpital's rule can be applied to this limit. (note: you can also use the Sandwich Theorem.)
 
Now the limit is in a usable form, which equals 1.

If the limit resulting from applying l'Hôpital's Rule is still one of the two mentioned indeterminates, we may apply the rule again (to the limit obtained), and again and again until a usable form is encountered.

Where does it come fromEdit

To obtain l'Hôpital's Rule for a limit of   which approaches   as   approaches  , we simply decompose both   and   in terms of their Taylor expansion (centered around  ). The independent terms of both expansions must be   (because both   and   approached  ), so if we divide both the   and   by   (or, equivalently, find their derivatives), our limit will stop being indeterminate.

It could be the case that the Taylor expansions of both the numerator and the denominator have a   as coefficient of the   term, thus yielding an indeterminate. This is the same case mentioned above where the trick was to repeat the process until a suitable limit was found.

The case of a limit which approaches   can be transformed to the case above by exchanging   with  , which obviously approaches  .

Calculus/Limits/Exercises