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Unformatted text preview: Section 1.3: The notion of limit Stewarts first definition: We write lim x a f ( x ) = L and say the limit of f(x ), as x approaches a , equals L if we can make the values of f ( x ) arbitrarily close to L by taking x to be sufficiently close (but not equal) to a . Why do we want to exclude x = a ? ... Because in many applications of limits (such as the definition of the derivative), we deal with functions where f is undefined at x = a (e.g., A ( h ) with h = 0, in the example on p. 24). [Draw picture from Stewart, using a red interval on the y- axis (with dashed horizontal lines) and a green interval on the x-axis (with dashed vertical lines).] lim x a f ( x ) = L means that for every red interval around L on the y-axis, there exists a green interval around a on the x- axis so that for every x in the green interval (except possibly x = a ), f ( x ) is defined and lies in the red interval. That is, the part of the graph y = f ( x ) that lies between the two green (vertical) lines lies between the two red (horizontal) lines (with the possible exception of ( a , f ( a )), which might be somewhere else or might not even be part...
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This note was uploaded on 02/13/2012 for the course MATH 141 taught by Professor Staff during the Fall '11 term at UMass Lowell.
- Fall '11