Rules of Limits

 

We have seen in the previous section that proving that the limit of a function at some value of x has a specific value is really a tedious task. Moreover, the method we used cannot find us the limit of the function, but only prove the value we know is true. In the following section, we will see how to find the limits of most of the functions we know. Let’s see the theorems we use for this object.

 

Theorem

The limit of the constant function , where c is a constant, at any value of x is equal to c. That is,

 

Theorem

The limit of the identity function  at some point is equal to the value of x at that point, or, in symbols:

 

Theorem

If we have two functions f and g whose limits exist at a number a, then,

1.    

2.    

3.    

4.     , in condition that n is any positive integer, and  when n is even.

 

 

That is a very important theorem to calculate the limits of many functions. It leads us also to the following corollaries.

 

Corollary

If  exists and k is a constant, then:

 

Corollary

If exists and n is any real number, then:

 

 

To find the limit of a function is now an easy task. We will show that in the following example.

 

Example

Find:

 

Solution

 

The following two corollaries are very helpful.

 

Corollary

Let f be the polynomial function defined by: , where n is a positive integer and  are constants. Then the limit of the function f at any number c is:

 

Corollary

Let f be a rational function, and c a number in its domain, then

 

NB

The limit of a function at some point has a unique value. No limit can have two values at the same time.

 

 

All what we saw before cannot yet be exploited except if the function is defined at the point of the limit. What about the limit when the function has an indeterminate value at some point? The following theorem is the answer.

 

Theorem

If we have two functions, f and g, such that , for all  except possibly at a, and  exists, we can say that

 

 

The use of this theorem is indicated by the following example.

 

Example

Let f be the function defined by:

 

Find:

 

Solution

Theorem

Let f be the function defined by:

such that a is a constant, and n is any real number. Then,

 

Corollary

If n, m and a are any real constants, then,

 

Example

Evaluate the limit:

 

Solution

 

To calculate the limits of some trigonometric functions, the following theorem and corollaries are needed.

 

Theorem

If x is an angle measured in radians, we say that

1.    

2.    

 

Corollary

If x is an angle measured in radians, and a and b are constants, then

1.     .

2.     .

 

Example

Calculate:

 

 

Solution