We call L the limit of f(x) as x

[cscott]

Definition

We call L the limit of f(x) as x approaches $\infty$ if for every number ε > 0 there exists a δ; such that whenever x > δ we have

$$\left| f(x) - L \right| < \epsilon$$

When this holds we write

$$\lim_{x \to \infty} f(x) = L$$

or

$$f(x) \to L \quad as \quad x \to \infty.$$

Similarly, we call L the limit of f(x) as x approaches $-\infty$ if for every number ε > 0, there exists a number δ such that whenever x < δ we have

$$\left| f(x) - L \right| < \epsilon$$

When this holds we write

$$\lim_{x \to -\infty} f(x) = L$$

or

$$f(x) \to L \quad as \quad x \to -\infty.$$

Notice the difference in these two definitions. For the limit of f(x) as x approaches $\infty$ we are interested in those x such that x > δ. For the limit of f(x) as x approaches $-\infty$ we are interested in those x such that x < δ.

What is δ that they keep referring to in this definition? They first bring it up as if it's some quantity I'm supposed to know... maybe I'm just crazy :)

 

[whozum]

They don't stand for antghing in particular in this situation, they are just variables (which could be replaced by any other ones) with the loose definitions given to them above. When I first did limits the epsilons and deltas threw me off too.

Do you understand how they're used in the limit definition?

 

[Galileo]

Suppose the limit exists, what it says is: Given $\epsilon>0$, there exists SOME number, such that whenever x is greater than that number, you have$|f(x)-L|<\epsilon$.
It's just convenient to name that number and they chose to name it $\delta$.

Example: $\lim_{x\to \infty} 1/x=0$
since given $\epsilon>0$ we can find a number $\delta$, such that $x>\delta \Rightarrow |1/x|<\epsilon$. Pick $\delta=1/\epsilon$, then for any $x>\delta=1/\epsilon$ we have $|1/x|<|1/\delta|=\epsilon$.

 

[Hurkyl]

How about this analogy:

For any real number N, there exists a real number M such that M > N.

Do you understand the meaning of N and M in this?

 

[JoshHolloway]

Differentiation Problem

I accidently posted in the wrong place, please forgive me.

 

[SteveRives]

What is δ that they keep referring to in this definition? They first bring it up as if it's some quantity I'm supposed to know... maybe I'm just crazy :)

Short answer: δ is the "run" portion of slope.

Limits are simple once you think of them as rise over run. Epsilon is rise, delta (δ) is run.

The classic assingments are usually of the form that the book will give you an epsilon, and you have to find the corresponding delta (at some given point on a curve). Given what I have just said, you can now think of it this way: they give you a rise, you give the corresponding run. They give you an e, you give them a δ -- where δ is "run".

I don't know of an easier way to think of it!

 

[dishuwen]

Epsilon-delta and limits

I have finally begun to understand the relationship between epsilon and delta. I can pretty much now use the delta-epsilon method to prove limits. The (ironic) problem, for me, now is that I'm having trouble finidng the limits! Can anyone offer a bit of advice or, maybe, direct me to a site on which I can practice some problems?

If anyone is having problems understanding the epsilon-delta relationship, I'll bge happy to try to explain my way of understanding it, if that would help.

 

[cscott]

Thanks for all your help, I understand it better now.

 

[SteveRives]

The (ironic) problem, for me, now is that I'm having trouble finidng the limits!

There are many things to say in answer to this; I will alert you to one of my favorites: L'Hopital's Rule. Learn the mechanics of it here:

http://www.math.hmc.edu/calculus/tutorials/lhopital/

Given what you have asked so far, I suspect the problems you are working on now will not require this knowledge, but there you have it...