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Unformatted text preview: LIMIT OF A FUNCTION Consider a function f ( x ) = x 2 + x + 1. As x approaches 1, f ( x ) gets closer to 3. In mathematics, we write it as follows lim x 1 f ( x ) = 3 (We say the limit of f ( x ), as x approaches 1, equals 3.) We can see this also from the graph of the function, as we move the point P ( a,a 2 + a + 1) on the graph closer to the point (1 , 3). Suppose that we have a slightly different function, f ( x ) = ( x 2 + x + 1 if x 6 = 1 2 if x = 1 . Even though f (1) = 2, the point P ( a,a 2 + a + 1) gets closer to the point (1 , 3) as we move the point P . Thus, again, lim x 1 f ( x ) = 3. Mathematically, the limit of a function is defined as follows: lim x a f ( x ) = L if we can make the values of f ( x ) arbitrarily close to L by taking x to be sufficiently close to a but not equal to a . We can see how it works through an example. Example 1. Guess the value of lim x 1 x 2 + 2 x- 3 x 2- 1 ....
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This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.
- Fall '09