Limits

Wednesday, 14 March 2018

3:07 PM

Definitions

 approaches L as x approaches infinity, the function  has limit .

 

We define the distance of  from the limit as the "".

This can be expressed as

When ,

When ,

When ,

 

;  when

 

Formal Definition

 

We can say that , if, for every  there exists a real number  such that if  then,

 

 

Proof

> Prove that

 

1. Get the limit L
2. Sub f(x) and L into the absolute value
3. Factorise L out (multiple by one (denominator/denominator))
4. Replace x with M
5. Rearrange M

sub g(x) and L into the absolute value,  factorise a 3 out, replace x with M and rearrange to find M in terms of epsilon.

From <https://moodle.telt.unsw.edu.au/mod/hsuforum/discuss.php?d=28773#/mod/hsuforum/discuss.php?d=28773#>

 

 

 

If , then

Consider

 

 

1. Find lim ()
2. Sub into

3. Set
4. Let
5. Solve for