Chapter 2 | Limits
205
pursuing a formal definition of infinite limits. To have lim x → a f ( x ) =+∞, we want the values of the function f ( x ) to get larger and larger as x approaches a . Instead of the requirement that | f ( x )− L | < ε for arbitrarily small ε when 0< | x − a | < δ for small enough δ , wewant f ( x ) > M for arbitrarily large positive M when 0< | x − a | < δ for small enough δ . Figure 2.43 illustrates this idea by showing the value of δ for successively larger values of M .
f ( x ) =+∞.
Figure 2.43 These graphs plot values of δ for M to show that lim x → a
Definition Let f ( x ) be defined for all x ≠ a in an open interval containing a . Then, we have an infinite limit lim x → a f ( x ) =+∞ if for every M >0, there exists δ >0 such that if 0< | x − a | < δ , then f ( x ) > M . Let f ( x ) be defined for all x ≠ a in an open interval containing a . Then, we have a negative infinite limit lim x → a f ( x ) =−∞ if for every M >0, there exists δ >0 such that if 0< | x − a | < δ , then f ( x ) <− M .
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