## 6.1. Limits

### Examples 6.1.3:

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

*{x*converging to

_{n}}*c*. Then we want to show that

*f(x*converges as well. So choose any

_{n})*> 0*and take a positive integer

*N*so large so that

*| x*for_{n}- c| < / m*n > N*

Then we have

for| f(x_{n}) - (m c + b) | = | m x_{n}+ b - m c - b | = m | x_{n}- c | < m / m =

*n > N*. But that means that the sequence

*f(x*converges to the number

_{n})*(m c + b)*.

2. Let *g(x) = [x]* and assume that *c* is an integer. Then take the
sequence

Then this sequence converges tox_{n}= c + (-1)^{n}/ n

*c*as n approaches infinity, but the sequence

*g(x*does not converge at all. Hence, the limit of

_{n})*g(x)*does not exist at any integer. On the other hand, if

*c*is not an integer, then the function

*g(x)*converges to the limit

*L = g(c)*. Can you prove it ?

3. Recalling our knowledge of topology, we remember that if *{x _{n}}*
is a sequence in a set

*converging to*

**D***c*, then

*c*is called an accumulation point of

*. Would it be correct to say that a function*

**D***f(x)*with domain

*converges to a limit*

**D***L*if for every accumulation point

*c*of

*the number*

**D***L*is also an accumulation point of the image of

*?*

**D**