epsilon-delta limit definition (1 Viewer)

[stupid idiot]

Ok, can someone explain to me how this definition guarantees the existence of the limit or what is the point of the definition. After many attempts to understand it, im still as confused as ever.

 

[Affinity]

.. definitions are definitions.. no explanations required

what's the whole point.. hmm
consider:

lim {x->c} f(x) = L
existence of limits say guarantees that as long as your x <B>close enough</B> to c,
then f(x) is as close to L as desired.

some examples of limits..
suppose everyone can get 100 in MATHXXXX if they spent enough time studying..
this is same as Lim {t->inifinity} g(t) = 100
where t is time spend on studying and g is the grade in maths as a function of t.

another example:
say you want to make some calculations with some measurements(with errors).. you would like to make sure that as long as your errors are small enough, then the results of your calculations based on the measurements will be close enough to the actual 'real value'
this condition is same (almost same besides some pathalogical counterexamples) as saying the limit exist for teh function, at the values concerned

 

[stupid idiot]

I get your intuition of close enough would guarentee an existence, but what i don't understand is the formal definition which states:

The limit of f(x) as x approaches c is L

Lim {x->c} f(x) = L if and only if,

for each e > 0, there exists a d > 0 such that

0 < |x - a| < d, then |f(x) - L| < e.

Can you show me how the limit will fail to exist if any of the above conditions are not met.

 

[Constip8edSkunk]

1st it should be 0 < |x - c| < d
you can interpret the definition geometrically: for x between c+d and c-d, the graph of f(x) would be within the area encased by L+e, L-e. If the conditions are not true then there will be some value of e (small enough) that the graph cannot be encased by any vertical strip bounded by c+d c-d, so the limit is wrong and/or does not exist

 

[jm1234567890]

I spent many hours before understanding it, then during lecture the lecturer says "we are going to ignore this section", grrrr

 

[Affinity]

show you how it will fail to exist? can't do that.. coz when people say the limit doesn't exist, they simply MEAN that one of the conditions/definitions are not met

 

[martin]

Ahh,
another question that I can pretend is revision for my analysis exam on thursday!

The definition of a limit of a function f:X->R is is that f converges to limit L as x approaches a if for every e>0 there exists a d>0 such that
|f(x) - L| < e whenever 0 < |x-a| < d

Basically this is saying that we can make f arbitrarily close to L if we make x arbitrarily close to a (but not a).

A useful way of thinking about it is as some kind of game: You give me an epsilon that you want the difference between f(x) and L to be and I give you back a delta so that whenever |x-a| < d, |f(x)-L| < e. The smaller the epsilon, the smaller the delta will need to be.

The point of the definition is that I find a function of epsilon to use as delta so I can always win the game.

We'll do a simple example,

Prove lim{x->2} 2x+4 = 8

so from the definition of the limit we need for each e>0 a d>0 such that |x-a| < d => |f(x)-L| < e

In this case we have L = 8 (Note you have to know, or guess, the limit to use this definition)

so to find d start with
|f(x) - L| < e
|2x+4 - 8| < e
|2x-4| < e
|x-2| < e/2

we need this to hold when
|x-a| < d
|x-2| < d

so take d = e/2

Now whenever you give me an epsilon I simply divide it by two and then by the reverse of the above chain of inequalities

|x-2| < e/2 => |2x+4 - 8| < e

cheers,
Martin

 

[stupid idiot]

Originally posted by martin
The point of the definition is that I find a function of epsilon to use as delta so I can always win the game.

Thats fun. I like it. Though im still in dark on how to connect the formal definition with an intuitive idea of a limit.

Maybe i should research the history of the definition.

 

[ezzy85]

i get the stuff martins saying but how can you prove the limit if you use the answer in your proof? like youre trying to prove lim{x->2} 2x+4 = 8 yet at the same time in your proof youre assuming L=8 is true. hows that prove anything if youre assuming its all true? im so lost...

 

[gman03]

Originally posted by Affinity
show you how it will fail to exist? can't do that.. coz when people say the limit doesn't exist, they simply MEAN that one of the conditions/definitions are not met

If the limit does not exist, there are other definitions we can use to prove it doesn't exist. e.g. proving it is approaching infinity etc.

 

[turtle_2468]

Basically you need a sense of whether or not it exists. (you can sometimes do this a la yr 12 by subbing in the number involved and seeing if it works, or l'hopital's/dividing by highest power of x etc etc...)
If doesn't exist: prove that.
If it does exist: Have an idea of what it is by above processes. Then you need to PROVE that it does indeed approach that limit (ie doesn't do dodgy stuff near the relevant point), and in order to do that you use epsilon-delta with a value of L (the value that you want to prove the limit to be)..