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## The Definiton of the Limit

### It's all Greek to me

Mathematicians, purely out of habit, use the greek symbols d (delta) and e (epsilon) when they mean really, really small real numbers. The idea is that these numbers can be as small as you like, as long as they are not exactly zero.

e and d, primarily because of their habitual use in the following definition of a limit, have a very special meaning to mathematicians. In some sense, they symbolize exactly what Analysis is all about (especially, its rigor and purity).

Anyway, don't be scared by them. They are no different, in principle, from the x which you've seen a million times. Remember, they just mean a really small number.

Here is the official definiton of the limit:

y is the limit of f(x) at a if for every e > 0 there exists a corresponding d > 0 so that if the distance between x and a is less than d (but not zero!) then the distance between f(x) and y is less than e.
and here is a (rigorous) interpretation:
y is the limit of the function f with input variable x near a given number called a if for any number, no matter how small, there exists a corresponding small number (that depends on the first small number) so that if the distance between x and a is less than the dependent small number (but not zero!) then the distance between f(x) and y is less than the first small number.

I suggest you draw a few pictures -- like the graph I drew -- and make sure you understand what each of the letters in the definition represents. Here are some further problems:

• Draw a picture of a function with a limit value different from its actual value at some input value.
• Can you draw a continuous function with a limit value different from its actual value? Does this suggest a good rigorous definiton of continuity to you?
• Why is it necessary to say that the distance between x and d cannot be zero?
• Can you draw a picture of a function which is not differentiable everywhere?
• Can you draw a picture of a differentiable function which is not continuous? What does this suggest? How would you prove it?

You now have some idea of what mathematicians mean by "rigorous"!

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