# How do epsilon-delta proofs prove the limit?

• Shaybay92
In summary: I recommend taking the time to get a good grasp of it, try thisIn summary, the proof of a limit provides evidence that the limit exists.
Shaybay92
I would really appreciate if someone could please explain to me how the final result of the proof of a limit actually PROVES the limit? http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preclimsoldirectory/PrecLimSol.html

I was particularly looking at Solution 2 on this website. I understand how |x-10| was factorized out etc. and therefore it can equal delta... but when it says that if delta = epsilon/3, |x-10| < epsilon/3 and therefore |f(x)-35| is less than epsilon... how is this true? I mean we don't even know what epsilon is?

how does this even prove that the limit is 35

Last edited:
I understand how |x-10| was factorized out etc. and therefore it can equal delta

Does it?

Do you understand the epsilon-delta argument. It is a two part formal method much used in mathematical analysis. Would this be your difficulty?

Hi Shaybay92!

(have a delta: δ and an epsilon: ε )
Shaybay92 said:
… I mean we don't even know what epsilon is?

ε is anything (> 0).

Continuity is proved if, for any ε, you can find a δ (which depends on that ε) which works.

ok but how does this proof tell us that there is a delta for any epsilon? i don't really see how if delta = cepsilon (c is some constant) then that is a proof...

Hi Shaybay92!

(what happened to that δ and ε i gave you? )
Shaybay92 said:
ok but how does this proof tell us that there is a delta for any epsilon? i don't really see how if delta = cepsilon (c is some constant) then that is a proof...

because, if δ is defined as cε, then for any ε, there is a δ !

so as long as x is less than a particular multiple of ε (cε) then |f(x)-L| < ε?

Shaybay92 said:
so as long as x is less than a particular multiple of ε (cε) then |f(x)-L| < ε?

That's right.

(Warning: although δ is usually a multiple of ε, somtimes it's something more awkward, like ε2 )

## What is the precise definition of a limit?

The precise definition of a limit is the mathematical concept that describes the behavior of a function as its input approaches a certain value. It is defined as the value that the function approaches as the input value gets closer and closer to the specified value.

## How is a limit symbolically expressed?

A limit is symbolically expressed using the notation "lim" followed by the input variable and the value it approaches. For example, "lim x→a" means the limit of the function as x approaches the value a.

## What is the significance of the limit in calculus?

The limit is a fundamental concept in calculus as it allows us to study the behavior of functions and solve complex problems involving rates of change, continuity, and infinite series. It is the basis for finding derivatives and integrals, which are crucial tools in calculus.

## What are the two types of limits?

The two types of limits are one-sided limits and two-sided limits. One-sided limits only consider the behavior of a function from one direction, while two-sided limits consider the behavior from both directions.

## How is the limit of a function evaluated?

The limit of a function can be evaluated using algebraic techniques, such as factoring and simplifying, or geometrically by analyzing the graph of the function. It can also be evaluated using numerical methods, such as plugging in values that are closer and closer to the specified value.

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