Chapter 4 | Applications of Derivatives
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Figure 4.47 For this function, the functional values approach infinity as x →±∞.
Definition (Informal) We say a function f has an infinite limit at infinity and write lim x →∞ f ( x ) =∞. if f ( x ) becomes arbitrarily large for x sufficiently large. We say a function has a negative infinite limit at infinity and write lim x →∞ f ( x ) =−∞. if f ( x ) <0 and | f ( x ) | becomes arbitrarily large for x sufficiently large. Similarly, we can define infinite limits as x →−∞. Formal Definitions Earlier, we used the terms arbitrarily close , arbitrarily large , and sufficiently large to define limits at infinity informally. Although these terms provide accurate descriptions of limits at infinity, they are not precise mathematically. Here are more formal definitions of limits at infinity. We then look at how to use these definitions to prove results involving limits at infinity. Definition (Formal) We say a function f has a limit at infinity , if there exists a real number L such that for all ε >0, there exists N >0 such that | f ( x )− L | < ε for all x > N . In that case, we write lim x →∞ f ( x ) = L (see Figure 4.48 ). We say a function f has a limit at negative infinity if there exists a real number L such that for all ε >0, there exists N <0 such that | f ( x )− L | < ε for all x < N . In that case, we write lim x →−∞ f ( x ) = L .
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