Limits

Definition : Precise definition of a Limit

For a function f defined in some open interval a (but not necessarily at a itself), we say , if given any (tiny) number > 0, there is another number d > 0 such that 0 < |x – a| < d guarantees that
|f(x) - L| < .

Definition of a Limit

Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement

  = L means that for each >0, there exists a >0 such that

if 0 < <

then < .

Limits can be found Numerically and Algebraically

While almost all limits can be found graphically, as we have been discussing, it is not always practical or necessary if the function is defined algebraically.

For instance, say we are given that. If we are looking for , instead of having to graphically search for the answer, we can find both the left- and right-hand limits by using tables. By choosing x values that get closer and closer to x = 3 from both sides, we can analyze the behavior of f(x).

Table 3.1

x	2.99	2.999	2.9999	3	3.0001	3.001	3.01
f(x)	5.99	5.999	5.9999	?	6.0001	6.001	6.01

Notice that when we chose values on either side of x = 3, they were values that were very close to x = 3. It seems that as x approaches 3 from either side, the function values are approaching 6. Therefore, it seems reasonable to conclude that

Limit Rules

If a, c, and n, are real numbers, then

1) (The limit of a constant real number is that number.)

2) where p(x) is any polynomial (The limit value of a polynomial is the function value at that point.)

3)

(The limit of the product of a constant and a function equals the constant times the limit of the function.)

4)

(The limit of the sum or difference of two functions equals the sum or difference of the limits of the functions.)

5)

(The limit of the product of two functions is the product of the limits of the functions.)

6)

(The limit of a quotient is the quotient of the limits of the numerator and denominator if the limit of the denominator is not zero.)

Example 8 Evaluate

1.  
2. 
3. 

Solution

1. (Rule 2)
2. (Rule 6)

                                  (Rule 2)

When you get 0/0 we have what is called an indeterminate form and we must try other techniques to determine the limit. In this case, factor both the numerator and denominator and cancel common terms to remove the zero in the denominator. Then, apply the limit rules to the simplified expression.

              (Factor)

                                     (Cancel common terms)

                                       = 3 + 3 = 6 (Rule 2)

1. (Rule 6)

                                 (Rules 1 and 2)

This is not defined and whenever you get a result of a non-zero number over zero, there are no common factors in the numerator and denominator which can be cancelled. Therefore, there is no way to rid the denominator of its zero term, meaning that the limit does not exist.

Example 9 Evaluate

Solution

are all also known as indeterminate forms. When this form occurs when finding limits at infinity (or negative infinity) with rational functions, divide every term in the numerator and denominator by the highest power of x in the denominator to determine the limit.

Since is the highest power of x in the denominator of our function, we have