Thread: How do we translate "lim" into English?

Hi.

Having looked at the derivation from first principles, I've understood the concept well, but I don't know what "lim" means.

Can anyone help?

Correct me if I'm wrong, but I think it's just short for limit.

But is it not a sentence like "The limit of" or something. E.g. When we are looking at differentiation from first principles, we have "lim h tends towards 0", how could we make this a full grammatical English sentence?

the limit of f as x approaches p is L.

Does "limit" here mean "maximum value"?

Limit means, roughly, the end of an inifinite series (although don't take that literally, as there is, by definition, no end to an infinite series).

How does "limit" relate then to differentiation from first principles i.e. f'(x) = the limit of f(x +h) - f(x) divided by h as h approaches 0 ?


I don't see the connection between f'(x) and "the end of an infinite series".

I've understood the concept well

i don't think you have.

It seems I may be frustrating some people, so apologies for that.

To explain my confusion in a different way, it seems to me that "value" would be a better word than "limit". As f'(x) = the value of f(x+h) - f(x) divided by h as h approaches 0.

I have always been taught that it is an abbreviation for the latin word "limes".
"Limes" is roughly translated to border or boundary, I think.

Originally Posted by russell_c_cook

It seems I may be frustrating some people, so apologies for that.

To explain my confusion in a different way, it seems to me that "value" would be a better word than "limit". As f'(x) = the value of f(x+h) - f(x) divided by h as h approaches 0.

No, that's what the equals sign is for. It's limit because when h is actually 0, you get 0/0 which is nonsense. Instead, you keep making h smaller and smaller and the expression continues to be meaningful however small you make it. Here you want to find the limit as h approaches 0, meaning where is it going.

Take f(x) = x. For any h except 0, (f(x+h)-f(x))/h is 1. At 0, it's 0/0. The limit as h goes to 0 is 1 though.

Write it out this way. Find the answer for h = 1, then h = 1/2, then 1/4, 1/8, 1/16 ...
The limit as h goes to 0 is the same as the the point of convergence of that series of numbers.

OK, thanks Magimaster.
Again apologies if I'm slow at getting the point, but I'm keen to learn and think its important to fully understand new terminology as it comes up.
The resources I have so far used have taken "limit" to have a self-evident meaning (to the point that it is introduced without any prior or subsequent explanation) that took a while for me to grasp.

Originally Posted by russell_c_cook

OK, thanks Magimaster.
Again apologies if I'm slow at getting the point, but I'm keen to learn and think its important to fully understand new terminology as it comes up.
The resources I have so far used have taken "limit" to have a self-evident meaning (to the point that it is introduced without any prior or subsequent explanation) that took a while for me to grasp.

If you are reading a book that takes the notion of a limit to be self-evident without any further definition, then throw it away and get another book.

I cannot imagine any author doing such a thing, but if that is indeed the case go to someone who knows what they are talking about.

Originally Posted by russell_c_cook

How does "limit" relate then to differentiation from first principles i.e. f'(x) = the limit of f(x +h) - f(x) divided by h as h approaches 0 ?


I don't see the connection between f'(x) and "the end of an infinite series".

You must look over the differentiation from first principles until you understand it better. Something I can say that may help you is this, however: a major principle in infinitesimal calculus is that infinitesimal quantities are often equal to whole values. For example, the sequence 0.9, 0.99, 0.999, 0.9999 and so on clearly tends to one, but it's also true that the infinitesimal 0.999... equals one, so we can actually use infinitesimals to represent limits.

Originally Posted by cook

Again apologies if I'm slow at getting the point,

It's a very difficult concept to get clear in one's mind. If it takes you several weeks and more, and many worked problems, that's not unusual.

I don't know what your resources are, but a good book and/or a good teacher are probably necessary. I doubt you can safely figure it out on your own, from the clues in some contextual uses of the thing.

Taking a guess at what you are up to: I would advise against attempting to learn calculus from a book until you are sure you have a handle on what a limit is.

I've always thought that it meant that it means for some real number e > 0 there is a natural number (1, 2, 3,...) m so that for any number greater than that natural number m, the absolute value of (x_m - the limit L) is less than the real number e.

That it sort of falls within a bound, which is made arbitrarily smaller and smaller.

When I see lim, I think of an asymptote ... as in what a hyperbola, exponential decay, etc approaches but never reaches.

So, I think of lim as: the value that the expression approaches (but never reaches) as the variable approaches its ultimate value. Thus, the function's "limit".

Some functions, such as x/(x²+x), are undefinable at the variable's ultimate value (in this case, x=0) due to division by zero, yet the function approaches a value of 1 as x approaches 0.

Other functions, such as (x+1/x)/x are definable but not computable due to the undefined value of the ultimate value (in this case, x=infinity), yet the function approaches a value of 1 as x approaches infinity.

but the limit can very well be reached. Example:
and, at that point, the function equals 1

Originally Posted by thyristor

I have always been taught that it is an abbreviation for the latin word "limes".
"Limes" is roughly translated to border or boundary, I think.

This is correct. However in software it's usually translated directly into function "limit" (e.g. in TI calculators). In Finnish we call it "Raja-arvo", which directly translates into Border value.