Stewart6e2_2 - MATH 191 Sections 1 and 3 Calculus I Fall...

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MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 2.2: The “L” Word: Limits! We need to be able to make mathematically precise the notion of “arbitrarily close, but not equal to.” (We encountered this notion just a little while back when we were trying to compute “instantaneous” velocities: we can find aver- age velocities over shorter and shorter time periods, but ultimately we end up dividing by 0 if we let our measured time period become as short as possible.) What does the trick? The notion of limit . Definition. We write lim x a f ( x ) = L if we can make f ( x ) as close as we want to to the number L by taking x sufficiently close, but not equal to, a . We often say “ f approaches L as x a .” In the space below, you should draw a few graphs, indicating the tricky nature of limits. First, draw a graph where the limit is equal to the value of the function at a : lim x a f ( x ) = f ( a ). Then, draw a graph showing that lim x a f ( x ) need not equal f ( a ). Finally, show that f ( a ) need not exist at all for the limit to exist:
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Examples.
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