2.6: Infinite Limits - Mathematics

Infinite Limits

Evaluating the limit of a function at a point or evaluating the limit of a function from the right and left at a point helps us to characterize the behavior of a function around a given value. As we shall see, we can also describe the behavior of functions that do not have finite limits.

We now turn our attention to (h(x)=1/(x−2)^2), the third and final function introduced at the beginning of this section (see Figure(c)). From its graph we see that as the values of x approach 2, the values of (h(x)=1/(x−2)^2) become larger and larger and, in fact, become infinite. Mathematically, we say that the limit of (h(x)) as x approaches 2 is positive infinity. Symbolically, we express this idea as

[lim_{x o 2}h(x)=+∞.]

More generally, we define infinite limits as follows:

Definitions: infinite limits

We define three types of infinite limits.

Infinite limits from the left: Let (f(x)) be a function defined at all values in an open interval of the form ((b,a)).

i. If the values of (f(x)) increase without bound as the values of x (where (x

ii. If the values of (f(x)) decrease without bound as the values of x (where (x

Infinite limits from the right: Let (f(x)) be a function defined at all values in an open interval of the form ((a,c)).

i. If the values of (f(x)) increase without bound as the values of x (where (x>a)) approach the number (a), then we say that the limit as x approaches a from the left is positive infinity and we write [lim_{x o a+}f(x)=+∞.]

ii. If the values of (f(x)) decrease without bound as the values of x (where (x>a)) approach the number (a), then we say that the limit as x approaches a from the left is negative infinity and we write [lim_{x o a+}f(x)=−∞.]

Two-sided infinite limit: Let (f(x)) be defined for all (x≠a) in an open interval containing (a)

i. If the values of (f(x)) increase without bound as the values of x (where (x≠a)) approach the number (a), then we say that the limit as x approaches a is positive infinity and we write [lim_{x o a} f(x)=+∞.]

ii. If the values of (f(x)) decrease without bound as the values of x (where (x≠a)) approach the number (a), then we say that the limit as x approaches a is negative infinity and we write [lim_{x o a}f(x)=−∞.]

It is important to understand that when we write statements such as (displaystyle lim_{x o a}f(x)=+∞) or (displaystyle lim_{x o a}f(x)=−∞) we are describing the behavior of the function, as we have just defined it. We are not asserting that a limit exists. For the limit of a function f(x) to exist at a, it must approach a real number L as x approaches a. That said, if, for example, (displaystyle lim_{x o a}f(x)=+∞), we always write (displaystyle lim_{x o a}f(x)=+∞) rather than (displaystyle lim_{x o a}f(x)) DNE.

Example (PageIndex{5}): Recognizing an Infinite Limit

Evaluate each of the following limits, if possible. Use a table of functional values and graph (f(x)=1/x) to confirm your conclusion.

1. (displaystyle lim_{x o 0−} frac{1}{x})
2. (displaystyle lim_{x o 0+} frac{1}{x})
3. ( displaystyle lim_{x o 0}frac{1}{x})

Solution

Begin by constructing a table of functional values.

(x)(frac{1}{x})(x)(frac{1}{x})
-0.1-100.110
-0.01-1000.01100
-0.001-10000.0011000
-0.0001-10,0000.000110,000
-0.00001-100,0000.00001100,000
-0.000001-1,000,0000.0000011,000,000

a. The values of (1/x) decrease without bound as (x) approaches 0 from the left. We conclude that

[lim_{x o 0−}frac{1}{x}=−∞. onumber]

b. The values of (1/x) increase without bound as (x) approaches 0 from the right. We conclude that

[lim_{x o 0+}frac{1}{x}=+∞. onumber]

c. Since (displaystyle lim_{x o 0−}frac{1}{x}=−∞) and (displaystyle lim_{x o 0+}frac{1}{x}=+∞) have different values, we conclude that

[lim_{x o 0}frac{1}{x}DNE. onumber]

The graph of (f(x)=1/x) in Figure (PageIndex{8}) confirms these conclusions.

Figure (PageIndex{8}): The graph of (f(x)=1/x) confirms that the limit as x approaches 0 does not exist.

Exercise (PageIndex{5})

Evaluate each of the following limits, if possible. Use a table of functional values and graph (f(x)=1/x^2) to confirm your conclusion.

1. (displaystyle lim_{x o 0−}frac{1}{x^2})
2. (displaystyle lim_{x o 0+}frac{1}{x^2})
3. (displaystyle lim_{x o 0}frac{1}{x^2})