2.1. Definition of limit

A limit is the value a function tends to take as the limit variable approaches a specified value. For example, the limit of the function

f (x) as x approaches the value of zero would be written as:

The value will come as close to zero as possible without actually becoming zero. The limit variable (x in example above) could approach some value other than zero. Most limits will have their limit variable approaching zero but that is not necessary and any other value will work just as well.

As an example, consider a function such as f (x) =3x²+2, and substitute x+x everywhere the variable x appears in the function to get the function f (x+x).

f (x)=3x²+2

f (x+x)=3(x+x)²+2=3(x²+2xx+(x)²)+2=3x²+6xx+3(x)²+2

Then the limit of the function f(x+x) as x approaches zero is written as:

=3x²+6x0+3(0)²+2=3x²+2

Obviously, in this example, the value of the limit is a function of x and a value can be computed if a value of x is specified. For x=2, the limit is 14, etc. You can find limit of a function as the limit variable approaches something other than zero. It could approach any other value within its range such as:

3x²+6x14+314²+2=3x²+84x+590

This method of evaluating a limit does not work for all functions. For example if:

then the limit of this function as x approaches the value of 4 cannot be computed by simply substituting for x. This function does not have a value for x=4 since we get a division of zero by zero by substituting

4 for x. This is defined as an indeterminent result since there is no real result defined for this type of division.

There are two possibilities here:

1. Factor the equation. In example above, the equation can be factored into:

Now we can substitute the value 4 for x to get the limit:

This works most of the times but not always.

2. Approach the value of the limit from both directions and calculate the value of the function. If you take values for x that approach the value 4, you will get answers other than divided by zero. And by taking values for x approaching 4 from both directions, you start to narrow down the value of limit. It becomes apparent that as x approaches the value 4 from either direction, the limit of the function approaches the value of 8.

A limit can also result in a value of infinity. For example, if the function is

then the limit of f (x) when x approaches zero would be:

obviously will approach infinity as x becomes closer and closer to zero. It cannot be directly calculated but can be determined by observation of the fact that the function would be a division by zero at the limit conditions, which is by definition equal to infinity.

Definition of limit: Let f (x) be defined for all x in some open interval containing the number a, with the possible exception that f (x) may not be defined at a. We will write