Stewart6e2_4

Stewart6e2_4 - MATH 191 Sections 1 and 3 Calculus I Fall...

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MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 2.4: Limits, Done Right! If we revisit the deﬁnition of “limit” that we got from Section 2.2, we notice that there are two places where we are left guided by our intuition, rather than mathematical precision: Intuitive deﬁnition. We say that lim x a f ( x ) = l if we can make f ( x ) arbitrarily close to L by choosing x suﬃciently close, but not equal to , a . We wish to make the two boldface phrases above a bit more mathematically precise. We do this with the following “redeﬁnition”: Precise deﬁnition. We say that lim x a f ( x ) = L if for any real number , there exists a real number , such that if 0 < | x - a | < , then | f ( x ) - L | < . That is, no matter how (arbitrarily) close we want to force f ( x ) to be to the value L (expressed precisely by the phrase “for any real number ± > 0”), we can draw near enough (namely, within δ > 0 of a ) to a in order that for such values of x , f ( x ) is close enough to (that is, within

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This note was uploaded on 04/08/2008 for the course MATH 191 taught by Professor Bahls during the Fall '07 term at UNC Asheville.

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Stewart6e2_4 - MATH 191 Sections 1 and 3 Calculus I Fall...

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