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Chapter 2 | Limits
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 2.12 (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 →2 h ( x ) =+∞. More generally, we define infinite limits as follows:
Definition 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 < a ) approach the number a , then we say that the limit as x approaches a from the left is positive infinity and we write (2.8) lim x → 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 (2.9) lim x → a − f ( 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 right is positive infinity and we write (2.10) lim x → 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 right is negative infinity and we write (2.11) lim x → 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 (2.12) lim x → 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 (2.13) lim x → a f ( x ) =−∞. It is important to understand that when we write statements such as lim x → 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, lim x → a f ( x ) =+∞, we always write lim x → a f ( x ) =+∞ rather than lim x → a f ( x ) DNE. f ( x ) =+∞ or lim x → a
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