## 6.1. Limits | IRA |

As the above easy example shows, things can be more complicated than anticipated. Therefore, we have to attack the problem more systematically. First, we need to define what we mean by 'limit of a function'.

Example 6.1.1:

- Consider the function
f, wheref(x) = 1ifx 0andf(x) = 2ifx > 0.

- The sequence
{ 1/n }converges to 0. What happens to the sequence{ f( 1/n ) }?- The sequence
{ 3 + (-1)is divergent. What happens to the sequence^{n}}{ f ( 3 + (-1)?^{n}) }- The sequence
{ (-1)converges to zero. What happens to the sequence^{n}/ n }{ f ( (-1)?^{n}/ n ) }

Definition 6.1.2: Limit of a Function (sequences version)A function fwith domainDin R converges to a limitLasxapproaches a numbercifis not empty and for any sequenceD - {c}{ xin_{n}that converges toD - {c}cthe sequence{ f ( xconverges to_{n}) }L.We write

f(x) = L

The above definition works quite well to show that a function is not continuous, because you only have to find one particular sequence whose images do not converge as a sequence. It is not a good definition, in general, to prove convergence of a function, because you will have to check every possible convergent sequence, and that is hard to do. We would therefore like another definition of convergence or limit of a function.

Examples 6.1.3:

- Apply this definition in these cases:

- Let
f(x) = m x + b. Then does the limit of that function exist at an arbitrary pointx?- Let
g(x) = [x], where[x]denotes the greatest integer less than or equal tox. Then does the limit ofgexist at an integer ? How about at numbers that are not integers ?- In the above definition, does
chave to be in the domainDof the function ? Iscin the closure(D) ? Do you know a name forcin terms of topology ?

Definition 6.1.4: Limit of a function (epsilon-delta Version)A function fwith domainDinR converges to a limitLasxapproaches a numbercclosure (D) if: given any> 0there exists a> 0such that:

- if
xandD| x - c | <then| f(x) - L | <

Regardless of which of the two definitions might be considered easier to use in a particular situation, the basic problem right now is that we have two different definitions for the same concept. We therefore have to show that both definitions are actually equivalent to each other.

Example 6.1.5:

In other words, both definitions of continuity are equivalent, and we can use which ever seems the easiest. Here are some basic properties of limits of functions.

Proposition 6.1.6: Equivalence of Definitions of LimitsIf fis any function with domainDinR, andcclosure(D) then the following are equivalent:

- For any sequence
{ xin_{n}}Dthat converges tocthe sequence{ f ( xconverges to_{n}) }L- given any
> 0there exists a> 0such ifxclosure(D) and| x - c | <then| f(x) - L | <

Sometimes a function may not have a limit using the above definitions, but when the domain of the function is restricted, then a limit exists. This leads to the concepts of one-sided limits.

Proposition 6.1.7: Properties for limits of Functions

- If
f(x)exists, the limit is unique.[ f(x) + g(x) ] = f(x) + g(x), provided thatf(x)andg(x)exist.[ f(x) g(x) ] = f(x) g(x), provided thatf(x)andg(x)exist.[ f(x) / g(x) ] = f(x) / g(x), provided thatf(x)andg(x)exist andg(x) # 0.

This is the formal definition of

Definition 6.1.8: One-Sided Limits of a FunctionIf fis a function with domainDandcclosure(D). Then:

fhas aleft-hand limitLatcif for every> 0there exists> 0such that ifxandDc - < x < cthen| f(x) - L | <. We writef(x) = L.fhas aright-hand limitLatcif for every> 0there exists> 0such that ifxandDc < x < c +then| f(x) - L | <. We writef(x) = L.

Now that we have some idea about limits of functions, we will move to the next question: if some sequence converges to

Proposition 6.1.9: Limits and One-Sided Limitsf(x) = Lif and only iff(x) = Landf(x) = L