Limits at Infinity

This is a different kind of limits, which is important especially in integration, but it may be somehow similar to the limits we already know.

First, Consider the function . This function is defined at all real numbers except at 0. Let’s make x represent largely increasing positive values as in the following table:

We can notice that the as the value of x increases, the function reaches values closer and closer to 0. In addition, we can make the function as close to 0 as we wish by assigning values for x, which are big enough. We can make the function differ from 0 by less than 0.001 by putting x greater than 1000. To make the function differ from 0 by less than 0.000001, we can simply put x > 1000000.

Moreover, when we make x take smaller and smaller negative values, the value of the function gets closer and closer to 0. That leads us to the definition of the limits at infinity, which is quite similar to that of the limit at a point.

Here we can write that  and .

Definition

Suppose we have a function f(x). We say that the limit of f as x increases without bound (or as x approaches infinity) is L, or in symbols:

 or ,

if the domain of f has no upper bound and if for each  there is a number M such that

,

where D is the domain of f.

Similarly, we say that the limit of f as x decreases without bound (or as x approaches negative infinity) is L, or in symbols:

,

if the domain of f has no lower bound and if for each  there is a number M such that

NB

The two symbols  and  mean the same.

The following theorem is used a lot in calculating the limits at infinity.

Theorem

Let n be any positive integer. Then:

NB

The theorems of addition, multiplication …etc. used in our previous subject of limits at a point is also true when dealing with limits at infinity. The following example clarifies that fact.

Example

Find:

Solution

Now we come back to the function with which we began our discussion, which is . Let’s try to find the right limit of the function at 0. We can see that the function is not approaching a certain number as x approaches 0 from the right, but it is increasing without bound. We say that . That leads us to the following definition.

Definition

We say that f(x) increases without bound as x approaches a from the right, or in symbols,

if and only if for every positive number M there is positive number , D is the domain of f, such that .

There are similar definitions for similar left limits and when f(x) decreases without bound.