Informal Definition Of Limits Assignment

While it is easy to define precisely in a few words what a square root is (?a is the positive number whose square is a) the definition of the limit of a function runs over several terse lines, and most people dont find it very enlightening when they first see it. (See 2.) So we postpone this for a while and fine tune our intuition for another page.

1.1. Definition of limit (1st attempt).

limx=a: f(x) = L
is read the limit of f(x) as x approaches a is L. It means that if you choose values of x which are close but not equal to a, then f(x) will be close to the value L; moreover, f(x) gets closer and closer to L as x gets closer and closer to a. The following alternative notation is sometimes used
f(x) =L as x=a;
(read f(x) approaches L as x approaches a or f(x) goes to L is x goes to a.)

1.2. Example

If f(x) = x + 3 then
limx=4: f(x) = 7,
is true, because if you substitute numbers x close to 4 in f(x) = x + 3 the result will be close to 7.

1.3. Example: substituting numbers to guess a limit.

What (if anything) is
limx=2: x² ? 2x /x² ? 4 ?
Here f(x) = (x² ? 2x)/(x2 ? 4) and a = 2.

We first try to substitute x = 2, but this leads to
f(2) = 2²-2 2/ 2²-4 = 0/0
which does not exist. Next we try to substitute values of x close but not equal to 2. Table 1 suggests that f(x) approaches 0.5.

1.4. Example: Substituting numbers can suggest the wrong answer.

The previous example shows that our first definition of limit is not very precise, because it says x close to a, but how close is close enough? Suppose we had taken the function

g(x) = 101 000x /100 000x + 1
and we had asked for the limit limx=0 g(x).

Then substitution of some small values of x could lead us to believe that the limit is 1.000 . . .. Only when you substitute even smaller values do you find that the limit is 0 (zero)!

See also problem 29.

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