Prove the statement using e and delta

Question

Prove the statement using e and s
lim (1-4x)=13
x→-3

Prove the statement using ε and s
please show steps and similar example

Answer 1

An epsilon-delta limit proof means that you have to prove the following:

 

For every ?>0, there exists a δ>0, such that for every x, if 0<|x−c|<δ then |f(x)−L|<?.

 

If that's true, then

 

lim f(x) = L
x->c

It sounds complicated, but what it means is that no matter how close x gets to c, f(x) will also stay "close" to L.

 

The key to the logic of these proofs is the statement "For every ?>0, there exists a δ>0..."  That means that you have to come up with a way to choose a δ based on any ? you are given.  If you can find a way to express δ in terms of ? that works for any ? you choose, then you have satisfied this "for every...there exists" criterion.

 

So for this specific limit, you have to prove: For every ?>0, there exists a δ>0, such that for every x, if 0<|x- -3|<δ then |(1-4x) −13|<?, or |-4x -12|<?.

 

The first step is to find a good expression for δ in terms of ?.  Work backwards from that last statement, the conclusion we will need to prove:

 

 

|-4x - 12| < ?

|-4||x + 3| < ?
4|x + 3| < ?

|x + 3| < ?/4

|x - -3| < ?/4

 

So, that suggests that given ?, we would want to pick δ = ?/4.  That is, for every ?, there exists a δ, which is equal to ?/4.

 

Now the proof.  Start with the δ statement ("if 0 <|x- -3| < δ") and try to derive the ? statement ("|(1 - 4x) − 13|< ?"

 

0 < |x- -3| < δ

Let δ = ?/4

0 < |x- -3| < ?/4

 

Multiply both sides by 4

0 < 4|x- -3|< ?

0 < |-4||x - -3| < ?

0 < |-4(x - -3)| < ?

0 < |-4x - 12)| < ?

0 < |(1-4x) −13| < ?

 

That's what we needed to prove.